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pressed in terms of one variable.

227. It is often convenient to express the coordinates

of any point on the curve in terms of one variable.

It is clear that the values

a 2a

mr 7n

always satisfy the equation to the curve.

Hence, for all values of m, the point

a 2a\

lies on the curve. By Art. 206, this m is equal to the

tangent of the angle v^hich the tangent at the point makes

v^^ith the axis.

The equation to the tangent at this point is

a

y = nix -\ â€” ,

and the normal is, by Art. 207, found to be

a

my + X = 2a + â€”-, .

in-

COORDINATES IN TERMS OF ONE VARIABLE. 199

228. The coordinates of the point could also be ex-

pressed in terms of the m of the normal at the point ; in

this case its coordinates are am?' and â€” 2am,

The equation of the tangent at the point (am^, â€” 2a7n)

is, by Art. 205,

mi/ + X + am^ â€” 0,

and the equation to the normal is

y â€” mx â€” 1am, â€” am?.

229. The simplest substitution (avoiding both nega-

tive signs and fractions) is

X = at2 and y = 2at.

These values satisfy the equation y^ = ^ax.

The equations to the tangent and normal at the point

{af, 2at) are, by Arts. 205 and 207,

ty = x + at^,

and y + tx= 2at + af.

The equation to the straight line joining

(atj^, 2at-^ and {at^, 2at^

is easily found to be

y {h + ^2) = 2x + 2at-f^.

The tangents at the points

{at^^ 2at^ and iat^^ 2at^

are t-^y =^x-\- at^,

and t.^y â€” x-\- at^.

The point of -intersection of these two tangents is clearly

{<X^1^2} ^(^1 + ^2)}-

The point whose coordinates are (a<^, 2aÂ£) may, for

brevity, be called the point " tT

In the following articles we shall prove some important

properties of the parabola making use of the above substi-

tution.

200 COORDINATE GEOMETRY.

230. If the tangents at P and Q meet in T, prove that

(1) TP and TQ subtend equal angles at the focus Sy

(2) ST^=SP.SQ,

and (3) the triangles SPT and STQ are similar.

Let P be the point {at^^, 2at^), and Q be the

point (at^^, 2at2), so that (Art. 229) T is the point

{atjt^, a(ii + t2)}-

2at

(1) The equation to SP is y= ^ ^ {x - a),

i. e. [t-^ -l)y-2t-^x + 2at-^ = 0.

The perpendicular, TU, from T on this

straight line

a{t ^^- 1) (fi + 1^) - 2^1 . ati tg + 2at^ ^ {t^^ - t^^t^) + (^i - ^

^^(fi-io)-

Similarly TC/' has the same numerical value.

The angles PST and QST are therefore equal.

(2) By Art. 202 we have SP=a{l + 1^^) and SQ = a(l + f.^).

Also ST^ = (af 1^2 -a)^ + a^{t^ + t^y-

= a^[tj;'t^^ + t^^ + t^^+l-\ = a^l + tj^){l + t^-^).

Hence ST^ = SP.SQ.

ST SO

(3) Since â€” ^ = ^, and the angles TSP and T^SQ are equal, the

oP ol

triangles SPT and STQ are similar, so that

Z /SQT^ z ^TP and Z ^rQ= z 5fPT.

231. 27ze area of the triangle formed by three points on a

parabola is twice the area of the triangle formed by the tangents at

these points.

Let the three points on the parabola be

{at-^, 2at-^, [at^, 2at^, and {at^^, 2at^.

The area of the triangle formed by these points, by Art. 25,

= i [at-^ {2af2 - Saig) + at^ {2at^ - 2at-^) + at^ {2at-^ - 2at^)'\

The intersections of the tangents at these points are (Art. 229)

the points

{at^t.^, a[t^ + t^\, {at^t^, a{t^-\-tT^], and [at^t^, a{t^-\-t^].

The area of the triangle formed by these three points

= \ {at^t^ {at^ - at^) + atj:-^ [at-^ - at^ + at-^t^ [at^-at^]

=W{t^-ts){t^-t^){t^-t^).

The first of these areas is double the second.

MISCELLANEOUS EXAMPLES. 201

232. The circle circumscribing the triangle formed by any three

tangents to a parabola passes through the focus.

Let P, Q, and R be the points at which the tangents are drawn

and let their coordinates be

{atj^, 2afj), (aig^ ^at^), and {at^^, 2at^).

As in Art. 229, the tangents at Q and R intersect in the point

{at^t^, a{t2 + t^)}.

Similarly, the other pairs of tangents meet at the points

{afgfi, ^(ig + ij)} and {at^t.^, a{tj^ + t2)}.

Let the equation to the circle be

x^ + y^ + 2gx + 2fy + c = (1).

Since it passes through the above three points, we have

a\%^ + a2 (<2 + t.^)^ + 2gat^t^ + 2/a (f g + fg) + c = (2),

aH^t^ + a2 {t^ + 1^'2 4- 2gat^t-^ + 2/a (tg + f^) + c = (3),

and aH-^t^ + a^ (Â«i + ^ ^ + 2gat-^t^ + 2fa{t^ + t^ + c = Q (4).

Subtracting (3) from (2) and dividing by a {t^ - t^), we have

a {fgS (fi + fa) + <i + ^3 + 2*3} + 2gt^ + 2/= 0.

Similarly, from (3) and (4), we have

Â«{Â«i'(*2 + Â«3) + Â«2 + *3 + 2Â«i} + 25rfi + 2/=0.

From these two equations we have

2g=-a{l-\- t^ts + t^t-i + tjt^) and 2f== -alt-^ + t.^ + t^- t^t^ts].

Substituting these values in (2), we obtain

c = a'^{t^ts + tstj^ + t-it2).

The equation to the circle is therefore

x^ + y^- ax (1 + Â£2*3 + *3^i + ^1^2) ~ ^^y ih + ^2 + ^3 ~ *i^2^3)

+ a- (t^t^ + t^t^ + tj^t^) = 0,

which clearly goes through the focus {a, 0).

233. If be any point on the axis and POP' be any chord

passing through 0, and if PM and P'M' be the ordinates of P and P',

prove that AM . AM' = AO\ and PM . P'M'= - 4a . AO.

Let O be the point {h, 0), and let P and P' be the points

(afj2, 2atj) and {at^, 2at^.

The equation to PP' is, by Art. 229,

{t^ + t-^y -2x = 2at^t^.

If this pass through the point [h, 0), we have

-2h = 2at-^t^,

202 COORDINATE GEOMETRY.

Hence AM. A M' =aU^. aU^ =a^.- = K^=A 0^-

and PM . PM' = 2a\ . 2atc^ . =4^2 ( - )=- 4a . AO.

icus, AO = a, and

Cor. If be the focus, ^0 = a, and we have

1

The points {at^, 2at-^ and ( â€” ^ , ) are therefore at the ends

of a focal chord.

234. To prove that the orthocentre of any triangle formed by

three tangents to a parabola lies on the directrix.

Let the equations to the three tangents be

y = nhx+â€” (1),

y=m^ + - (2),

nt

and y = mgX-\ (3).

m.

The point of intersection of (2) and (3) is found, by solving them,

to be

The equation to the straight line through this point perpendicular

to (1) is (Art. 69)

y-a{â€” + â€”) = \ X ~ ,

X rl 1 a ~\

I.e. ?/+â€” =a â€” + â€” + (4).

Similarly, the equation to the straight line through the intersection

of (3) and (1) perpendicular to (2) is

X ( \ 1 d \

y + â€” =a _ + _+ , 5)

and the equation to the straight line through the intersection of (1)

and (2) perpendicular to (3) is

X f \ \ ci \

?/ + â€” = a â€” +â€” + 6.

wig \^\ ^2 ni-^m^m^J

The point which is common to the straight lines (4), (5), and (6),

EXAMPLES. ONE VARIABLE. 203

i.e. the orthocentre of the triangle, is easily seen to be the point

whose coordinates are

/111 1

x=-a^ y = a[â€”'-\ ! +

and this point lies on the directrix.

EXAMPLES. XXVIII.

1. If w be the angle which a focal chord of a parabola makes with

the axis, prove that the length of the chord is 4a cosec'-^ w and that the

perpendicular on it from the vertex is a sin w.

2. A point on a parabola, the foot of the perpendicular from it

upon the directrix, and the focus are the vertices of an equilateral

triangle. Prove that the focal distance of the point is equal to the

latus rectum.

3. Prove that the semi-latus-rectum is a harmonic mean between

the segments of any focal chord.

4. If T be any point on the tangent at any point P of a parabola,

and if TL be perpendicular to the focal radius SP and TN be perpen-

dicular to the directrix, prove that SL = TN.

Hence obtain a geometrical construction for the pair of tangents

drawn to the parabola from any point T.

5. Prove that on the axis of any parabola there is a certain point

K which has the property that, if a chord PQ of the parabola be drawn

through it, then

1 1

PK^'^'QK^

is the same for all positions of the chord.

6. The normal at the point (at-^, 2atj) meets the parabola again

in the point {aU^, 2at2) ; prove that

2

H

7. A chord is a normal to a parabola and is inclined at an angle

d to the axis ; prove that the area of the triangle formed by it and

the tangents at its extremities is 4a'^ sec^ 6 cosec^ 0.

8. If PQ be a normal chord of the parabola and if S be the focus,

prove that the locus of the centroid of the triangle SPQ is the curve

36a2/2 (3a; - 5a) - 81^^= 128a'^.

9. Prove that the length of the intercept on the normal at the

point {at^, 2at) made by the circle which is described on the focal

distance of the given point as diameter is a f^/l + 1^.

204 COORDINATE GEOMETRY. [EXS.

10. Prove that the area of the triangle formed by the normals to

the parabola at the points {at^, Satj), {at^, 2at^ and [at^y Satg) is

\ (Â«2 - h) (Â«3 - h) ih - h) ih + *2 + Â«3)'.

11. Prove that the normal chord at the point whose ordinate

is equal to its abscissa subtends a right angle at the focus.

12. A chord of a parabola passes through a point on the axis

(outside the parabola) whose distance from the vertex is half the

latus rectum ; prove that the normals at its extremities meet on the

curve.

13. The normal at a point P of a parabola meets the curve

again in Q, and T is the pole of PQ; shew that T lies on the diameter

passing through the other end of the focal chord passing through P,

and that PT is bisected by the directrix.

14. If from the vertex of a parabola a pair of chords be drawn at

right angles to one another and with these chords as adjacent sides a

rectangle be made, prove that the locus of the further angle of the

rectangle is the parabola

?/2 = 4a (a; -8a).

15. A series of chords is drawn so that their projections on a

straight line which is inclined at an angle a to the axis are all of

constant length c ; prove that the locus of their middle point is the

curve

{y^-iax) {y qoq a -\-2a Bin of + a^c^ = 0.

16. Prove that the locus of the poles of chords which subtend a

right angle at a fixed point (/i, /c) is

ax^ - %2 + (4a'^ + 2ali) x - 2ahj + a {h^ + A;-) = 0.

17. Prove that the locus of the middle points of all tangents

drawn from points on the directrix to the parabola is

y^{2x + a) = a{dx + a)-.

18. Prove that the orthocentres of the triangles formed by three

tangents and the corresponding three normals to a parabola are

equidistant from the axis.

19. T is the pole of the chord PQ ; prove that the perpendiculars

from P, T, and Q upon any tangent to the parabola are in geometrical

progression.

20. If '^1 and r^ be the lengths of radii vectores of the parabola

which are drawn at right angles to one another from the vertex, prove

that

rji^r2^=16a2{ri^ + r2^).

21. A parabola touches the sides of a triangle ABC in the points

D, E, and F respectively ; if DE and DF cut the diameter through the

point A inb and c respectively, prove that Bb and Cc are parallel.

XXVIII.] EXAMPLES. ONE VARIABLE. 205

22. Prove that all circles described on focal radii as diameters

touch the directrix of the curve, and that all circles on focal radii as

diameters touch the tangent at the vertex,

23. -A- circle is described on a focal chord as diameter ; if m be the

tangent of the inclination of the chord to the axis, prove that the

equation to the circle is

\ m^J m

24. LOL' and il/Oilf'are two chords of a parabola passing through

a point on its axis. Prove that the radical axis of the circles

described on LL' and MM' as diameters passes through the vertex of

the parabola.

25. -A- circle and a parabola intersect in four points; shew that the

algebraic sum of the ordinates of the four points is zero.

Shew also that the line joining one pair of these four points and

the line joining the other pair are equally inclined to the axis.

26. Circles are drawn through the vertex of the parabola to cut

the parabola orthogonally at the other point of intersection. Prove

that the locus of the centres of the circles is the curve

2i/2 (22/2 + a;2 - Viax) = ax {%x - 4a)2.

27. Prove that the equation to the circle passing through the

points {at^, 2,at^ and (2a<2^, 2at^ and the intersection of the tan-

gents to the parabola at these points is

ic2 + 2/2 - ax [(Â«! + Â«2)^ + 2] - aij [t^ + 1^ (1 - t-^ t^) + a2 1^ t^ (2 - 1-^ t^ = 0.

28. TP and TQ are tangents to the parabola and the normals at P

and Q meet at a point R on the curve ; prove that the centre of the

circle circumscribing the triangle TPQ Kes on the parabola

2y^ = a{x â€” a).

29. Through the vertex A of the parabola ?/2 = 4aa; two chords AP

and AQ are drawn, and the circles on AP and ^Q as diameters

intersect in R. Prove that, if 6-^, 6^,^ and be the angles made with

the axis by the tangents at P and Q and by AR, then

cot ^i + cot ^2 + 2 tan0 = O.

30. A- parabola is drawn such that each vertex of a given triangle

is the pole of the opposite side ; shew that the focus of the parabola

lies on the nine-point circle of the triangle, and that the orthocentre of

the triangle formed by joining the middle points of the sides lies on

the directrix.

CHAPTER XL

THE PARABOLA {continued).

[On a first reading of this Chapter, the student may, with

advantage, omit from Art. 239 to the end.]

Some examples of Loci connected with the

Parabola.

235. Ex. 1. Find the locus of the intersection of tangents to the

â€¢parabola y^ = 4iax, the angle hetioeen them being always a given angle a.

The straight line y ^mx -] â€” is always a tangent to the parabola.

If it pass through the point T {h, h) we

have

m^h-mk + a = (1).

If mj and Wg be the roots of this equation -Â»-

we have (by Art, 2) (/ik)

h

and

a

.(2),

â€¢(3),

and the equations to TP and TQ are then

a

and y â€” nux -^ â€” - .

1 m^

yâ€”miX + â€” Â»^Â« yâ€”ni,^.

Hence, by Art. 66, we have

m^ - m^ _ ijjm-^ + m^)^ - ^m^ m^

tana

1 + m^m^

1 + WljWg

/

k^ _4a

a + h

h

, by (2) and (3).

THE PARABOLA. LOCI. 207

.-. k^-4:ah = {a + h)^t&n^a.

Hence the coordinates of the point T always satisfy the equation

7/2 - 4kax = {a + x)^ tan^ a.

We shall find in a later chapter that this curve is a hyperbola.

As a particular case let the tangents intersect at right angles, so

that m^^= - 1.

From (3) we then have h= -a, so that in this case the point T lies

on the straight line x= -a, which is the directrix.

Hence the locus of the point of intersection of tangents, which cut

at right angles, is the directrix.

Ex. 2. Prove that the locus of the poles of chords which are normal

to the parabola y^ = 4:ax is the curve

2/2(a; + 2a) + 4a3 = 0.

Let PQ be a chord which is normal at P. Its equation is then

y = mx-2am-am^ (1).

Let the tangents at P and Q. intersect in T, whose coordinates are

h and k, so that we require the locus of T.

Since PQ is the polar of the point {h, k) its equation is

yk=2a{x + h) (2).

Now the equations (1) and (2) represent the same straight line, so

that they must be equivalent. Hence

m = â€” , and - 2am - am^ = â€”jâ€” .

Eliminating m, i. e. substituting the value of m from the first of

these equations in the second, we have

4a2 8a^ _ 2ah

~T~'W~~k''

i.e. ^ k^{h + 2a} + 4a^=0,

The locus of the point T is therefore

2/2(a; + 2a) + 4a3=0.

Ex. 3. Find the locus of the middle points of chords of a parabola

which subtend a right angle at the vertex, and prove that these chords all

pass through a fixed point on the axis of the curve.

208

COORDINATE GEOMETRY.

First Method. Let PQ be any sucli chord, and let its equation be

y = mx + c (1).

The lines joining the vertex with the

points of intersection of this straight line y

with the parabola

y^=^ax (2),

are given by the equation A

y^c = 4ax {y - mx). (Art. 122)

These straight lines are at right angles if

c + 4am=:0. (Art. Ill)

Substituting this value of c in (1), the

equation to FQ is

y = m {x - 4a)

This straight line cuts the axis of a; at a constant distance 4a from

the vertex, i.e. AA' = 4a.

If the middle point oi PQ be {h, k) we have, by Art. 220,

2a

(3).

li=:

(4).

.(5).

m

Also the point {h, k) lies on (3), so that we have

k = m{h-4a)

If between (4) and (5) we eliminate m, we have

kJ-^{h-4a),

i.e. k^=1a{h-4ia),

so that {h, k) always lies on the parabola

2/2 = 2a (a; -4a).

This is a parabola one half the size of the original, and whose

vertex is at the point A' through which all the chords pass.

Second Method. Let P be the point (afj^, 2at^ and Q be the point

{at^, 2aio).

The tangents of the inclinations of AP and AQ io the axis are

â€” and â€” .

1 2

Since AP and AQ are at right angles, therefore

*1 *2

.(6).

i.e. tih= -4

As in Art. 229 the equation to PQ is

{ty^ + t^y=:2x + 2at^t^ (7).

THE PARABOLA. LOCI. 209

This meets the axis of ic at a distance -atit2,i.e., by (6), 4a, from

the origin.

Also, {h, k) being the middle point of PQ, we have

and 2k = 2a{tj^ + t2).

Hence k- - 2ah = a^ (f^ + 1^)^ - a^ {tj^ + 1^^)

= 2a\t2= -8a2,

so that the locus of (h, k) is, as before, the parabola

y^ â€” 2a{x- 4a).

Tbird Method. The equation to the chord which is bisected at

the point {h, k) is, by Art. 221,

k{y-k) = 2a{x-h),

i. e. ky - 2ax=k^ - 2ah (8).

As in Art. 122 the equation to the straight lines joining its points

of intersection with the parabola to the vertex is

{k^ - 2ah) 2/2 = iax {ky - 2ax).

These lines are at right angles if

{k^-2ah) + 8a^ = 0.

Hence the locus as before.

Also the equation (8) becomes

ky - 2ax = - 8a^.

This straight line always goes through the point (4a, 0).

EXAMPLES. XXIX.

From an external point P tangents are drawn to the parabola ; find

the equation to the locus of P when these tangents make angles 6^ and

$2 with the axis, such that

1. tan dj^ + tan 6^ is constant ( = 6).

2. tan 6^ tan d^ is constant { = c).

3. cot 6^ + cot $2 is constant { = d).

4. di + ^2 is constant ( = 2a).

5 . tan^ 6-^ + tan^ ^g is constant ( = X) .

6. cos ^1 cos $2 is constant ( = /u),

L. 14

210 COORDINATE GEOMETRY. [ExS.

7. Two tangents to a parabola meet at an angle of 45Â° ; prove that

the locus of their point of intersection is the curve

y^ - \.ax = {x + of.

If they meet at an angle of 60Â°, prove that the locus is

2/2-3a;2-10aic-3a2^0.

8. A pair of tangents are drawn which are equally inclined to a

straight line whose inclination to the axis is a ; prove that the locus

of their point of intersection is the straight line

y = {x-a) tan 2a.

9. Prove that the locus of the point of intersection of two tangents

which intercept a given distance 4c on the tangent at the vertex is an

equal parabola.

10. Shew that the locus of the point of intersection of two tangents,

which with the tangent at the vertex form a triangle of constant area

c^, is the curve x^ [y^ - 4:ax)=4:C^a^.

11. If the normals at P and Q meet on the parabola, prove that

the point of intersection of the tangents at P and Q lies either on a

certain straight line, which is parallel to the tangent at the vertex, or

on the curve whose equation is y^ {x + 2a) + 4a^ = 0.

12. Two tangents to a parabola intercept on a fixed tangent

segments whose product is constant ; prove that the locus of their

point of intersection is a straight line.

13. Shew that the locus of the poles of chords which subtend a

constant angle a at the vertex is the curve

(x + A.af=4: cot^ a (t/^ - 4aa;).

14. In the preceding question if the constant angle be a right angle

the locus is a straight line perpendicular to the axis.

15. A point P is such that the straight line drawn through it

perpendicular to its polar with respect to the parabola y^=4:ax touches

the parabola x'^ = Aby. Prove that its locus is the straight line

2ax + by + 4:a^=0.

16. Two equal parabolas, A and B, have the same vertex and axis

but have their concavities turned in opposite directions ; prove that

the locus of poles with respect to B of tangents to A is the parabola A.

17. Prove that the locus of the poles of tangents to the parabola

y^â€”Aax with respect to the circle x^ + y'^=2ax is the circle x^ + y^=ax.

18. Shew the locus of the poles of tangents to the parabola

y^â€”Aax with respect to the parabola y^=4bx is the parabola

y'^= â€” X.

XXIX.] THREE NORMALS FROM ANY POINT.

211

Find the locus of the middle points of chords of the parabola

which

19. pass through the focus.

20. pass through the fixed point (/?., k).

21. are normal to the curve.

22. subtend a constant angle a at the vertex.

23. are of given length I.

24. are such that the normals at their extremities meet on the

parabola.

25. Through each point of the straight line x = niy + h is drawn

the chord of the parabola y^=4iax which is bisected at the point;

prove that it always touches the parahola

{y - 2am)^=8d {x - h).

26. Two parabolas have the same axis and tangents are drawn to

the second from points on the first ; prove that the locus of the middle

points of the chords of contact with the second parabola all lie on a

fixed parabola.

27. Prove that the locus of the feet of the perpendiculars drawn

from the vertex of the parabola upon chords, which subtend an angle

of 45Â° at the vertex, is the curve

r2 - 24ar cos 6 + 16a^ cos 29 = 0.

236. To prove that, in general, three normals can be

drawn from any point to the parahola and that the algebraic

sum, of the ordinates of the feet of these three normals is

zero.

The straight line

y â€” mx â€” 2am â€” amP

is, by Art. 208, a normal to the

parabola at the points whose coordi-

nates are

arn^ and â€” 2am (2).

If this normal passes through

the fixed point 0, vi^hose coordinates

are h and k, we have

k = mh â€” 2am, â€” am,\

%.e.

arn? + (2a â€” h) m + k

(1)

212 COORDINATE GEOMETRY.

This equation, being of the third degree, has three

roots, real or imaginary. Corresponding to each of these

roots, we have, on substitution in (1), the equation to a

normal which passes through the point 0.

Hence three normals, real or imaginary, pass through

any point 0.

If m^, iiu, and m^ be the roots of the equation (3), we

have

m^ + m.^ + m^ = 0.

If the ordinates of the feet of these normals be 2/^, y^,

and 2/3, we then have, by (2),

2/1 + 2/2 + 2/3 = - 2o^ {m^ + ^2 + m^) = 0.

Hence the second part of the proposition.

We shall find, in a subsequent chapter, that, for certain

positions of the point 0, all three normals are real ; for

other positions of 0, one normal only will be real, and the

other two imaginary.

237. Ex. Find the locus of a point which is such that (a) two of

the normals drawn from it to the parabola are at right angles,

(j8) the three normals through it cut the axis in points whose distances

from, the vertex are in arithjnetical progression.

Any normal is y=mx-2am-am^, and this passes through the

point {h, k), if

am^ + {2a-h)m+k = (1).

If then nil , m^, and m^ be the roots, we have, by Art. 2,

m-y + m^ + m^ = , (2) ,

m^m^ + m^mi + mim^= , (3),

k

and mim2m^= â€” (4).

(a) If two of the normals, say m^ and m^^ , be at right angles, we

k

have wiim2=-l, and hence, from (4), m^â€”-.

k

The quantity - is therefore a root of (1) and hence, by substitution,

Of

we have

-+{2a-h)~ + k = 0,

a^ ^ a

i.e. k^=a{h-3a).

THREE NORMALS. EXAMPLES. 213

The locus of the point (/^, k) is therefore the parabola y'^ = a{x- 3a)

whose vertex is the point (3a, 0) and whose latus rectum is one-quarter

that of the given parabola.

The student should draw the figure of both parabolas.

(/3) The normal y = mx - 2am - am^ meets the axis of a; at a point

whose distance from the vertex is 2a + aw^. The conditions of the

question then give

(2a + awij^) + (2a + am^^) = 2 (2a + am^^) ,

i.e. m-^ + m.^ = 2m<^ (5).

If we eliminate m^, m^, and m^ from the equations (2), (3), (4),

and (5) we shall have a relation between h and k.

From (2) and (3), we have

= vi^m.^ + m^ (wii + m^) = m^m^ - m^^ (6).

Also, (5) and (2) give

2m2^ = {v\ + wig)^ - Im^m.^ â€” m.^ - Im-^^ ,

i.e. m^ + 2m^m^ =0 (7).

Solving (6) and (7), we have

2a - 7t , â€ž ^ 2a-h

vi.mo= â€”z â€” , and ot^-^ - 2 x â€”r â€” .

^ â€¢* 3a -^ 3a

Substituting these values in (4), we have

2 a- fe / 2a-h_ k

3a V ~Sa^ ~ ~ a '

i.e. 27afc2 = 2(7i-2a)3,

so that the required locus is

27ai/2=2(a;-2a)3.

238. Ex. If the normals at three points P, Q, and R meet in a

point and S be the focus, prove that SP . SQ . SR = a . S0\

As in the previous question we know that the normals at the

points (ayn-^^, -2amj), {ain.^,-2am2^ and {am.^,-2am^) meet in the

point (A, k) if

mj + ^2 + m3 = (1),

2a ~ h

m^vi.^ + m^m-^ + m^m^ â€” â€” (2) ,

and mnW.,??io = â€” (3).

'â– - " a ^ '

By Art. 202 we have

SP=a{l+m^), SQ = a{l + m^-), and SR = a{l + m^^y

214 COORDINATE GEOMETRY.

Hence ^P-^Q-^^ ^ ^1 _^ ^^^2) (i + ^^^2) (i + ,,,^.2)

= 1 + (Wj^ + m^^ + m^) + [m^m^ + m^m^ + m^m^) + m^m^m.^.

Also, from (1) and (2),' we have

m^ + 7n2^ + Wg^ = (wi + 7712 + wis)^ - 2 (?Â»oW3 + ma?^! + m^mo)

_ h-2a

a

and

= (^y,by (l)and(2).

â€ž SP.SQ.SR ^ li-2a fli-2ay Â¥â–

227. It is often convenient to express the coordinates

of any point on the curve in terms of one variable.

It is clear that the values

a 2a

mr 7n

always satisfy the equation to the curve.

Hence, for all values of m, the point

a 2a\

lies on the curve. By Art. 206, this m is equal to the

tangent of the angle v^hich the tangent at the point makes

v^^ith the axis.

The equation to the tangent at this point is

a

y = nix -\ â€” ,

and the normal is, by Art. 207, found to be

a

my + X = 2a + â€”-, .

in-

COORDINATES IN TERMS OF ONE VARIABLE. 199

228. The coordinates of the point could also be ex-

pressed in terms of the m of the normal at the point ; in

this case its coordinates are am?' and â€” 2am,

The equation of the tangent at the point (am^, â€” 2a7n)

is, by Art. 205,

mi/ + X + am^ â€” 0,

and the equation to the normal is

y â€” mx â€” 1am, â€” am?.

229. The simplest substitution (avoiding both nega-

tive signs and fractions) is

X = at2 and y = 2at.

These values satisfy the equation y^ = ^ax.

The equations to the tangent and normal at the point

{af, 2at) are, by Arts. 205 and 207,

ty = x + at^,

and y + tx= 2at + af.

The equation to the straight line joining

(atj^, 2at-^ and {at^, 2at^

is easily found to be

y {h + ^2) = 2x + 2at-f^.

The tangents at the points

{at^^ 2at^ and iat^^ 2at^

are t-^y =^x-\- at^,

and t.^y â€” x-\- at^.

The point of -intersection of these two tangents is clearly

{<X^1^2} ^(^1 + ^2)}-

The point whose coordinates are (a<^, 2aÂ£) may, for

brevity, be called the point " tT

In the following articles we shall prove some important

properties of the parabola making use of the above substi-

tution.

200 COORDINATE GEOMETRY.

230. If the tangents at P and Q meet in T, prove that

(1) TP and TQ subtend equal angles at the focus Sy

(2) ST^=SP.SQ,

and (3) the triangles SPT and STQ are similar.

Let P be the point {at^^, 2at^), and Q be the

point (at^^, 2at2), so that (Art. 229) T is the point

{atjt^, a(ii + t2)}-

2at

(1) The equation to SP is y= ^ ^ {x - a),

i. e. [t-^ -l)y-2t-^x + 2at-^ = 0.

The perpendicular, TU, from T on this

straight line

a{t ^^- 1) (fi + 1^) - 2^1 . ati tg + 2at^ ^ {t^^ - t^^t^) + (^i - ^

^^(fi-io)-

Similarly TC/' has the same numerical value.

The angles PST and QST are therefore equal.

(2) By Art. 202 we have SP=a{l + 1^^) and SQ = a(l + f.^).

Also ST^ = (af 1^2 -a)^ + a^{t^ + t^y-

= a^[tj;'t^^ + t^^ + t^^+l-\ = a^l + tj^){l + t^-^).

Hence ST^ = SP.SQ.

ST SO

(3) Since â€” ^ = ^, and the angles TSP and T^SQ are equal, the

oP ol

triangles SPT and STQ are similar, so that

Z /SQT^ z ^TP and Z ^rQ= z 5fPT.

231. 27ze area of the triangle formed by three points on a

parabola is twice the area of the triangle formed by the tangents at

these points.

Let the three points on the parabola be

{at-^, 2at-^, [at^, 2at^, and {at^^, 2at^.

The area of the triangle formed by these points, by Art. 25,

= i [at-^ {2af2 - Saig) + at^ {2at^ - 2at-^) + at^ {2at-^ - 2at^)'\

The intersections of the tangents at these points are (Art. 229)

the points

{at^t.^, a[t^ + t^\, {at^t^, a{t^-\-tT^], and [at^t^, a{t^-\-t^].

The area of the triangle formed by these three points

= \ {at^t^ {at^ - at^) + atj:-^ [at-^ - at^ + at-^t^ [at^-at^]

=W{t^-ts){t^-t^){t^-t^).

The first of these areas is double the second.

MISCELLANEOUS EXAMPLES. 201

232. The circle circumscribing the triangle formed by any three

tangents to a parabola passes through the focus.

Let P, Q, and R be the points at which the tangents are drawn

and let their coordinates be

{atj^, 2afj), (aig^ ^at^), and {at^^, 2at^).

As in Art. 229, the tangents at Q and R intersect in the point

{at^t^, a{t2 + t^)}.

Similarly, the other pairs of tangents meet at the points

{afgfi, ^(ig + ij)} and {at^t.^, a{tj^ + t2)}.

Let the equation to the circle be

x^ + y^ + 2gx + 2fy + c = (1).

Since it passes through the above three points, we have

a\%^ + a2 (<2 + t.^)^ + 2gat^t^ + 2/a (f g + fg) + c = (2),

aH^t^ + a2 {t^ + 1^'2 4- 2gat^t-^ + 2/a (tg + f^) + c = (3),

and aH-^t^ + a^ (Â«i + ^ ^ + 2gat-^t^ + 2fa{t^ + t^ + c = Q (4).

Subtracting (3) from (2) and dividing by a {t^ - t^), we have

a {fgS (fi + fa) + <i + ^3 + 2*3} + 2gt^ + 2/= 0.

Similarly, from (3) and (4), we have

Â«{Â«i'(*2 + Â«3) + Â«2 + *3 + 2Â«i} + 25rfi + 2/=0.

From these two equations we have

2g=-a{l-\- t^ts + t^t-i + tjt^) and 2f== -alt-^ + t.^ + t^- t^t^ts].

Substituting these values in (2), we obtain

c = a'^{t^ts + tstj^ + t-it2).

The equation to the circle is therefore

x^ + y^- ax (1 + Â£2*3 + *3^i + ^1^2) ~ ^^y ih + ^2 + ^3 ~ *i^2^3)

+ a- (t^t^ + t^t^ + tj^t^) = 0,

which clearly goes through the focus {a, 0).

233. If be any point on the axis and POP' be any chord

passing through 0, and if PM and P'M' be the ordinates of P and P',

prove that AM . AM' = AO\ and PM . P'M'= - 4a . AO.

Let O be the point {h, 0), and let P and P' be the points

(afj2, 2atj) and {at^, 2at^.

The equation to PP' is, by Art. 229,

{t^ + t-^y -2x = 2at^t^.

If this pass through the point [h, 0), we have

-2h = 2at-^t^,

202 COORDINATE GEOMETRY.

Hence AM. A M' =aU^. aU^ =a^.- = K^=A 0^-

and PM . PM' = 2a\ . 2atc^ . =4^2 ( - )=- 4a . AO.

icus, AO = a, and

Cor. If be the focus, ^0 = a, and we have

1

The points {at^, 2at-^ and ( â€” ^ , ) are therefore at the ends

of a focal chord.

234. To prove that the orthocentre of any triangle formed by

three tangents to a parabola lies on the directrix.

Let the equations to the three tangents be

y = nhx+â€” (1),

y=m^ + - (2),

nt

and y = mgX-\ (3).

m.

The point of intersection of (2) and (3) is found, by solving them,

to be

The equation to the straight line through this point perpendicular

to (1) is (Art. 69)

y-a{â€” + â€”) = \ X ~ ,

X rl 1 a ~\

I.e. ?/+â€” =a â€” + â€” + (4).

Similarly, the equation to the straight line through the intersection

of (3) and (1) perpendicular to (2) is

X ( \ 1 d \

y + â€” =a _ + _+ , 5)

and the equation to the straight line through the intersection of (1)

and (2) perpendicular to (3) is

X f \ \ ci \

?/ + â€” = a â€” +â€” + 6.

wig \^\ ^2 ni-^m^m^J

The point which is common to the straight lines (4), (5), and (6),

EXAMPLES. ONE VARIABLE. 203

i.e. the orthocentre of the triangle, is easily seen to be the point

whose coordinates are

/111 1

x=-a^ y = a[â€”'-\ ! +

and this point lies on the directrix.

EXAMPLES. XXVIII.

1. If w be the angle which a focal chord of a parabola makes with

the axis, prove that the length of the chord is 4a cosec'-^ w and that the

perpendicular on it from the vertex is a sin w.

2. A point on a parabola, the foot of the perpendicular from it

upon the directrix, and the focus are the vertices of an equilateral

triangle. Prove that the focal distance of the point is equal to the

latus rectum.

3. Prove that the semi-latus-rectum is a harmonic mean between

the segments of any focal chord.

4. If T be any point on the tangent at any point P of a parabola,

and if TL be perpendicular to the focal radius SP and TN be perpen-

dicular to the directrix, prove that SL = TN.

Hence obtain a geometrical construction for the pair of tangents

drawn to the parabola from any point T.

5. Prove that on the axis of any parabola there is a certain point

K which has the property that, if a chord PQ of the parabola be drawn

through it, then

1 1

PK^'^'QK^

is the same for all positions of the chord.

6. The normal at the point (at-^, 2atj) meets the parabola again

in the point {aU^, 2at2) ; prove that

2

H

7. A chord is a normal to a parabola and is inclined at an angle

d to the axis ; prove that the area of the triangle formed by it and

the tangents at its extremities is 4a'^ sec^ 6 cosec^ 0.

8. If PQ be a normal chord of the parabola and if S be the focus,

prove that the locus of the centroid of the triangle SPQ is the curve

36a2/2 (3a; - 5a) - 81^^= 128a'^.

9. Prove that the length of the intercept on the normal at the

point {at^, 2at) made by the circle which is described on the focal

distance of the given point as diameter is a f^/l + 1^.

204 COORDINATE GEOMETRY. [EXS.

10. Prove that the area of the triangle formed by the normals to

the parabola at the points {at^, Satj), {at^, 2at^ and [at^y Satg) is

\ (Â«2 - h) (Â«3 - h) ih - h) ih + *2 + Â«3)'.

11. Prove that the normal chord at the point whose ordinate

is equal to its abscissa subtends a right angle at the focus.

12. A chord of a parabola passes through a point on the axis

(outside the parabola) whose distance from the vertex is half the

latus rectum ; prove that the normals at its extremities meet on the

curve.

13. The normal at a point P of a parabola meets the curve

again in Q, and T is the pole of PQ; shew that T lies on the diameter

passing through the other end of the focal chord passing through P,

and that PT is bisected by the directrix.

14. If from the vertex of a parabola a pair of chords be drawn at

right angles to one another and with these chords as adjacent sides a

rectangle be made, prove that the locus of the further angle of the

rectangle is the parabola

?/2 = 4a (a; -8a).

15. A series of chords is drawn so that their projections on a

straight line which is inclined at an angle a to the axis are all of

constant length c ; prove that the locus of their middle point is the

curve

{y^-iax) {y qoq a -\-2a Bin of + a^c^ = 0.

16. Prove that the locus of the poles of chords which subtend a

right angle at a fixed point (/i, /c) is

ax^ - %2 + (4a'^ + 2ali) x - 2ahj + a {h^ + A;-) = 0.

17. Prove that the locus of the middle points of all tangents

drawn from points on the directrix to the parabola is

y^{2x + a) = a{dx + a)-.

18. Prove that the orthocentres of the triangles formed by three

tangents and the corresponding three normals to a parabola are

equidistant from the axis.

19. T is the pole of the chord PQ ; prove that the perpendiculars

from P, T, and Q upon any tangent to the parabola are in geometrical

progression.

20. If '^1 and r^ be the lengths of radii vectores of the parabola

which are drawn at right angles to one another from the vertex, prove

that

rji^r2^=16a2{ri^ + r2^).

21. A parabola touches the sides of a triangle ABC in the points

D, E, and F respectively ; if DE and DF cut the diameter through the

point A inb and c respectively, prove that Bb and Cc are parallel.

XXVIII.] EXAMPLES. ONE VARIABLE. 205

22. Prove that all circles described on focal radii as diameters

touch the directrix of the curve, and that all circles on focal radii as

diameters touch the tangent at the vertex,

23. -A- circle is described on a focal chord as diameter ; if m be the

tangent of the inclination of the chord to the axis, prove that the

equation to the circle is

\ m^J m

24. LOL' and il/Oilf'are two chords of a parabola passing through

a point on its axis. Prove that the radical axis of the circles

described on LL' and MM' as diameters passes through the vertex of

the parabola.

25. -A- circle and a parabola intersect in four points; shew that the

algebraic sum of the ordinates of the four points is zero.

Shew also that the line joining one pair of these four points and

the line joining the other pair are equally inclined to the axis.

26. Circles are drawn through the vertex of the parabola to cut

the parabola orthogonally at the other point of intersection. Prove

that the locus of the centres of the circles is the curve

2i/2 (22/2 + a;2 - Viax) = ax {%x - 4a)2.

27. Prove that the equation to the circle passing through the

points {at^, 2,at^ and (2a<2^, 2at^ and the intersection of the tan-

gents to the parabola at these points is

ic2 + 2/2 - ax [(Â«! + Â«2)^ + 2] - aij [t^ + 1^ (1 - t-^ t^) + a2 1^ t^ (2 - 1-^ t^ = 0.

28. TP and TQ are tangents to the parabola and the normals at P

and Q meet at a point R on the curve ; prove that the centre of the

circle circumscribing the triangle TPQ Kes on the parabola

2y^ = a{x â€” a).

29. Through the vertex A of the parabola ?/2 = 4aa; two chords AP

and AQ are drawn, and the circles on AP and ^Q as diameters

intersect in R. Prove that, if 6-^, 6^,^ and be the angles made with

the axis by the tangents at P and Q and by AR, then

cot ^i + cot ^2 + 2 tan0 = O.

30. A- parabola is drawn such that each vertex of a given triangle

is the pole of the opposite side ; shew that the focus of the parabola

lies on the nine-point circle of the triangle, and that the orthocentre of

the triangle formed by joining the middle points of the sides lies on

the directrix.

CHAPTER XL

THE PARABOLA {continued).

[On a first reading of this Chapter, the student may, with

advantage, omit from Art. 239 to the end.]

Some examples of Loci connected with the

Parabola.

235. Ex. 1. Find the locus of the intersection of tangents to the

â€¢parabola y^ = 4iax, the angle hetioeen them being always a given angle a.

The straight line y ^mx -] â€” is always a tangent to the parabola.

If it pass through the point T {h, h) we

have

m^h-mk + a = (1).

If mj and Wg be the roots of this equation -Â»-

we have (by Art, 2) (/ik)

h

and

a

.(2),

â€¢(3),

and the equations to TP and TQ are then

a

and y â€” nux -^ â€” - .

1 m^

yâ€”miX + â€” Â»^Â« yâ€”ni,^.

Hence, by Art. 66, we have

m^ - m^ _ ijjm-^ + m^)^ - ^m^ m^

tana

1 + m^m^

1 + WljWg

/

k^ _4a

a + h

h

, by (2) and (3).

THE PARABOLA. LOCI. 207

.-. k^-4:ah = {a + h)^t&n^a.

Hence the coordinates of the point T always satisfy the equation

7/2 - 4kax = {a + x)^ tan^ a.

We shall find in a later chapter that this curve is a hyperbola.

As a particular case let the tangents intersect at right angles, so

that m^^= - 1.

From (3) we then have h= -a, so that in this case the point T lies

on the straight line x= -a, which is the directrix.

Hence the locus of the point of intersection of tangents, which cut

at right angles, is the directrix.

Ex. 2. Prove that the locus of the poles of chords which are normal

to the parabola y^ = 4:ax is the curve

2/2(a; + 2a) + 4a3 = 0.

Let PQ be a chord which is normal at P. Its equation is then

y = mx-2am-am^ (1).

Let the tangents at P and Q. intersect in T, whose coordinates are

h and k, so that we require the locus of T.

Since PQ is the polar of the point {h, k) its equation is

yk=2a{x + h) (2).

Now the equations (1) and (2) represent the same straight line, so

that they must be equivalent. Hence

m = â€” , and - 2am - am^ = â€”jâ€” .

Eliminating m, i. e. substituting the value of m from the first of

these equations in the second, we have

4a2 8a^ _ 2ah

~T~'W~~k''

i.e. ^ k^{h + 2a} + 4a^=0,

The locus of the point T is therefore

2/2(a; + 2a) + 4a3=0.

Ex. 3. Find the locus of the middle points of chords of a parabola

which subtend a right angle at the vertex, and prove that these chords all

pass through a fixed point on the axis of the curve.

208

COORDINATE GEOMETRY.

First Method. Let PQ be any sucli chord, and let its equation be

y = mx + c (1).

The lines joining the vertex with the

points of intersection of this straight line y

with the parabola

y^=^ax (2),

are given by the equation A

y^c = 4ax {y - mx). (Art. 122)

These straight lines are at right angles if

c + 4am=:0. (Art. Ill)

Substituting this value of c in (1), the

equation to FQ is

y = m {x - 4a)

This straight line cuts the axis of a; at a constant distance 4a from

the vertex, i.e. AA' = 4a.

If the middle point oi PQ be {h, k) we have, by Art. 220,

2a

(3).

li=:

(4).

.(5).

m

Also the point {h, k) lies on (3), so that we have

k = m{h-4a)

If between (4) and (5) we eliminate m, we have

kJ-^{h-4a),

i.e. k^=1a{h-4ia),

so that {h, k) always lies on the parabola

2/2 = 2a (a; -4a).

This is a parabola one half the size of the original, and whose

vertex is at the point A' through which all the chords pass.

Second Method. Let P be the point (afj^, 2at^ and Q be the point

{at^, 2aio).

The tangents of the inclinations of AP and AQ io the axis are

â€” and â€” .

1 2

Since AP and AQ are at right angles, therefore

*1 *2

.(6).

i.e. tih= -4

As in Art. 229 the equation to PQ is

{ty^ + t^y=:2x + 2at^t^ (7).

THE PARABOLA. LOCI. 209

This meets the axis of ic at a distance -atit2,i.e., by (6), 4a, from

the origin.

Also, {h, k) being the middle point of PQ, we have

and 2k = 2a{tj^ + t2).

Hence k- - 2ah = a^ (f^ + 1^)^ - a^ {tj^ + 1^^)

= 2a\t2= -8a2,

so that the locus of (h, k) is, as before, the parabola

y^ â€” 2a{x- 4a).

Tbird Method. The equation to the chord which is bisected at

the point {h, k) is, by Art. 221,

k{y-k) = 2a{x-h),

i. e. ky - 2ax=k^ - 2ah (8).

As in Art. 122 the equation to the straight lines joining its points

of intersection with the parabola to the vertex is

{k^ - 2ah) 2/2 = iax {ky - 2ax).

These lines are at right angles if

{k^-2ah) + 8a^ = 0.

Hence the locus as before.

Also the equation (8) becomes

ky - 2ax = - 8a^.

This straight line always goes through the point (4a, 0).

EXAMPLES. XXIX.

From an external point P tangents are drawn to the parabola ; find

the equation to the locus of P when these tangents make angles 6^ and

$2 with the axis, such that

1. tan dj^ + tan 6^ is constant ( = 6).

2. tan 6^ tan d^ is constant { = c).

3. cot 6^ + cot $2 is constant { = d).

4. di + ^2 is constant ( = 2a).

5 . tan^ 6-^ + tan^ ^g is constant ( = X) .

6. cos ^1 cos $2 is constant ( = /u),

L. 14

210 COORDINATE GEOMETRY. [ExS.

7. Two tangents to a parabola meet at an angle of 45Â° ; prove that

the locus of their point of intersection is the curve

y^ - \.ax = {x + of.

If they meet at an angle of 60Â°, prove that the locus is

2/2-3a;2-10aic-3a2^0.

8. A pair of tangents are drawn which are equally inclined to a

straight line whose inclination to the axis is a ; prove that the locus

of their point of intersection is the straight line

y = {x-a) tan 2a.

9. Prove that the locus of the point of intersection of two tangents

which intercept a given distance 4c on the tangent at the vertex is an

equal parabola.

10. Shew that the locus of the point of intersection of two tangents,

which with the tangent at the vertex form a triangle of constant area

c^, is the curve x^ [y^ - 4:ax)=4:C^a^.

11. If the normals at P and Q meet on the parabola, prove that

the point of intersection of the tangents at P and Q lies either on a

certain straight line, which is parallel to the tangent at the vertex, or

on the curve whose equation is y^ {x + 2a) + 4a^ = 0.

12. Two tangents to a parabola intercept on a fixed tangent

segments whose product is constant ; prove that the locus of their

point of intersection is a straight line.

13. Shew that the locus of the poles of chords which subtend a

constant angle a at the vertex is the curve

(x + A.af=4: cot^ a (t/^ - 4aa;).

14. In the preceding question if the constant angle be a right angle

the locus is a straight line perpendicular to the axis.

15. A point P is such that the straight line drawn through it

perpendicular to its polar with respect to the parabola y^=4:ax touches

the parabola x'^ = Aby. Prove that its locus is the straight line

2ax + by + 4:a^=0.

16. Two equal parabolas, A and B, have the same vertex and axis

but have their concavities turned in opposite directions ; prove that

the locus of poles with respect to B of tangents to A is the parabola A.

17. Prove that the locus of the poles of tangents to the parabola

y^â€”Aax with respect to the circle x^ + y'^=2ax is the circle x^ + y^=ax.

18. Shew the locus of the poles of tangents to the parabola

y^â€”Aax with respect to the parabola y^=4bx is the parabola

y'^= â€” X.

XXIX.] THREE NORMALS FROM ANY POINT.

211

Find the locus of the middle points of chords of the parabola

which

19. pass through the focus.

20. pass through the fixed point (/?., k).

21. are normal to the curve.

22. subtend a constant angle a at the vertex.

23. are of given length I.

24. are such that the normals at their extremities meet on the

parabola.

25. Through each point of the straight line x = niy + h is drawn

the chord of the parabola y^=4iax which is bisected at the point;

prove that it always touches the parahola

{y - 2am)^=8d {x - h).

26. Two parabolas have the same axis and tangents are drawn to

the second from points on the first ; prove that the locus of the middle

points of the chords of contact with the second parabola all lie on a

fixed parabola.

27. Prove that the locus of the feet of the perpendiculars drawn

from the vertex of the parabola upon chords, which subtend an angle

of 45Â° at the vertex, is the curve

r2 - 24ar cos 6 + 16a^ cos 29 = 0.

236. To prove that, in general, three normals can be

drawn from any point to the parahola and that the algebraic

sum, of the ordinates of the feet of these three normals is

zero.

The straight line

y â€” mx â€” 2am â€” amP

is, by Art. 208, a normal to the

parabola at the points whose coordi-

nates are

arn^ and â€” 2am (2).

If this normal passes through

the fixed point 0, vi^hose coordinates

are h and k, we have

k = mh â€” 2am, â€” am,\

%.e.

arn? + (2a â€” h) m + k

(1)

212 COORDINATE GEOMETRY.

This equation, being of the third degree, has three

roots, real or imaginary. Corresponding to each of these

roots, we have, on substitution in (1), the equation to a

normal which passes through the point 0.

Hence three normals, real or imaginary, pass through

any point 0.

If m^, iiu, and m^ be the roots of the equation (3), we

have

m^ + m.^ + m^ = 0.

If the ordinates of the feet of these normals be 2/^, y^,

and 2/3, we then have, by (2),

2/1 + 2/2 + 2/3 = - 2o^ {m^ + ^2 + m^) = 0.

Hence the second part of the proposition.

We shall find, in a subsequent chapter, that, for certain

positions of the point 0, all three normals are real ; for

other positions of 0, one normal only will be real, and the

other two imaginary.

237. Ex. Find the locus of a point which is such that (a) two of

the normals drawn from it to the parabola are at right angles,

(j8) the three normals through it cut the axis in points whose distances

from, the vertex are in arithjnetical progression.

Any normal is y=mx-2am-am^, and this passes through the

point {h, k), if

am^ + {2a-h)m+k = (1).

If then nil , m^, and m^ be the roots, we have, by Art. 2,

m-y + m^ + m^ = , (2) ,

m^m^ + m^mi + mim^= , (3),

k

and mim2m^= â€” (4).

(a) If two of the normals, say m^ and m^^ , be at right angles, we

k

have wiim2=-l, and hence, from (4), m^â€”-.

k

The quantity - is therefore a root of (1) and hence, by substitution,

Of

we have

-+{2a-h)~ + k = 0,

a^ ^ a

i.e. k^=a{h-3a).

THREE NORMALS. EXAMPLES. 213

The locus of the point (/^, k) is therefore the parabola y'^ = a{x- 3a)

whose vertex is the point (3a, 0) and whose latus rectum is one-quarter

that of the given parabola.

The student should draw the figure of both parabolas.

(/3) The normal y = mx - 2am - am^ meets the axis of a; at a point

whose distance from the vertex is 2a + aw^. The conditions of the

question then give

(2a + awij^) + (2a + am^^) = 2 (2a + am^^) ,

i.e. m-^ + m.^ = 2m<^ (5).

If we eliminate m^, m^, and m^ from the equations (2), (3), (4),

and (5) we shall have a relation between h and k.

From (2) and (3), we have

= vi^m.^ + m^ (wii + m^) = m^m^ - m^^ (6).

Also, (5) and (2) give

2m2^ = {v\ + wig)^ - Im^m.^ â€” m.^ - Im-^^ ,

i.e. m^ + 2m^m^ =0 (7).

Solving (6) and (7), we have

2a - 7t , â€ž ^ 2a-h

vi.mo= â€”z â€” , and ot^-^ - 2 x â€”r â€” .

^ â€¢* 3a -^ 3a

Substituting these values in (4), we have

2 a- fe / 2a-h_ k

3a V ~Sa^ ~ ~ a '

i.e. 27afc2 = 2(7i-2a)3,

so that the required locus is

27ai/2=2(a;-2a)3.

238. Ex. If the normals at three points P, Q, and R meet in a

point and S be the focus, prove that SP . SQ . SR = a . S0\

As in the previous question we know that the normals at the

points (ayn-^^, -2amj), {ain.^,-2am2^ and {am.^,-2am^) meet in the

point (A, k) if

mj + ^2 + m3 = (1),

2a ~ h

m^vi.^ + m^m-^ + m^m^ â€” â€” (2) ,

and mnW.,??io = â€” (3).

'â– - " a ^ '

By Art. 202 we have

SP=a{l+m^), SQ = a{l + m^-), and SR = a{l + m^^y

214 COORDINATE GEOMETRY.

Hence ^P-^Q-^^ ^ ^1 _^ ^^^2) (i + ^^^2) (i + ,,,^.2)

= 1 + (Wj^ + m^^ + m^) + [m^m^ + m^m^ + m^m^) + m^m^m.^.

Also, from (1) and (2),' we have

m^ + 7n2^ + Wg^ = (wi + 7712 + wis)^ - 2 (?Â»oW3 + ma?^! + m^mo)

_ h-2a

a

and

= (^y,by (l)and(2).

â€ž SP.SQ.SR ^ li-2a fli-2ay Â¥â–

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