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y'^^4ax' (2),

and y"^^4.aa/' (3).

TANGENT AT ANY POINT OF A PARABOLA. 181

Hence, by subtraction, we have

2/"' - y"' ^ 4a (x" - x),

i.e.

{y"-y'){y" + y') = 4.a{x"-x'),

and hence

y" -y 4o&

X â€”X y -^ y

Substituting this value in equation (1), we have, as

the equation to any secant PQ^

4a

i.e. y {y + y") = ^ax + y'y" + y^ â€” 4:ax

â€” iax + y'y" (4).

To obtain the equation of the tangent at (x\ y') we take

Q indefinitely close to P, and hence, in the limit, put y" = y .

The equation (4) then becomes

^yy â€” y"^ + 4aa; = 4^ax + 4aa;',

i.e. yy' = 2a(x + x').

Cor. It will be noted that the equation to the tangent

is obtained from the equation to the curve by the rule of

Art. 152.

Exs. The equation to the tangent at the point {2, -4) of the

parabola y^=Qx is

2/(-4) = 4(a; + 2),

i.e. x-\-y + 1 = 0.

The equation to the tangent at the point ( â€” 2 Â» â€” ) of the parabola

y^=4iax is

2a . f a\

y . â€” = 2a[x + ~] ,

d

I.e. ii = mx-\ â€” .

m

206. To find the condition that the straight line

y = mx + c (1)

rtiay touch the parabola y^ â€” 4:ax (2).

The abscisses of the points in which the straight line (1)

meets the curve (2) are as in Art. 203, given by the equation

rn^x^ + 2x {mo - 2a) + c^ = (3).

182 COORDINATE GEOMETRY.

The line (1) will touch (2) if it meet it in two points

which are indefinitely close to one another, i.e. in two

points which ultimately coincide.

The roots of equation (3) must therefore be equal.

The condition for this is

4 {mc â€” 2ay â€” iirrc^

i.e. ct? - amc â€” 0,

a

so that G = â€” .

m

Substituting this value of c in (1), we have as the

equation to a tangent,

a

y = mx + â€” .

m

In this equation tyi is the tangent of th6 angle which

the tangent makes with the axis of x.

The foregoing proposition may also be obtained from the equation

of Art. 205.

For equation (4) of that article may be written

la lax' ,^.

y=â€”x+^ (1).

y y

In this equation put â€” , =m, i.e. y =â€” ,

^ y m

, y'^ a , 2ax' a

and hence x = -r- = ^, , and â€” â€” = â€” .

4a m^ y m

a

The equation (1) then becomes y = mx-] â€” .

Also it is the tangent at the point {x', y'), i.e. (â€”^, â€” j .

207. Equation to the normal at (x, y'). The required

normal is the straight line which passes through the point

{x\ y') and is perpendicular to the tangent, i.e. to the

straight line

2a . ,.

yz=zâ€”-{x + x).

Its equation is therefore

y â€” y = rri' {x â€” x'),

2fi 11

where m' x â€” rr - 1 i.e. ^i' = -^ (Art. 69.)

y 2a

NORMAL TO A PARABOLA.

183

,(1).

and the equation to the normal is

y-y'=^(x-x')

208. To exj^ress the equation of the normal in the form

y = mx â€” 2a7n â€” am^.

In equation (1) of the last article put

2a

= m, %.e. y

- â€” ^.

am.

/2

Hence

, y 2

X =^ = aiir.

4a

The normal is therefore

y + 2am = m {x

am

%

t.e.

y = mx â€” 2am â€” am^

and it is a normal at the point (am^, â€” 2am) of the curve.

In this equation m is the tangent of the angle which

the normal makes with the axis. It must be carefully

distinguished from the m of Art. 206 which is the tangent

of the angle which the tangent makes with the axis. The

" m" of this article is - 1 divided by the " m'' of Art. 206.

209. Subtangent and Subnormal. Def. If

the tangent and normal at any point P of a conic section

meet the axis in T and G respectively and PN be the

ordinate at P, then NT is called the Subtangent and NG the

Subnormal of P.

To find the length of the subtangent and suhnorm^al.

If P be the point {x\ y') the equation to TP is, by

Art. 205,

yy â€”2a{x-\- x) (1).

To obtain the length of AT^ we

have to find the point where this

straight line meets the axis of tc,

i.e. we put 2/ = 0in (1) and we

have

x=^-x (2).

Hence AT=AN,

184 COORDINATE GEOMETRY.

[The negative sign in equation (2) shews that T and

N always lie on opposite sides of the vertex -4.]

Hence the subtangent iV^= 2J^iV = twice the abscissa

of the point P.

Since TFG is a right-angled triangle, we have (Euc. vi. 8)

FN'^^TN.NG.

Hence the subnormal NG

_ PiP _ PN^

The subnormal is therefore constant for all points on

the parabola and is equal to the semi-latus rectum.

210. Ex. 1. If a chord which is normal to the parabola at one

end subtend a right angle at the vertex, prove that it is inclined at an

angle tan~^ ^J2 to the axis.

The equation to any chord which is normal is

y = mx â€” 2am - am^,

i.e. mx-y = 2am+am^.

The parabola is y^ â€” 4:ax.

The straight lines joining the origin to the intersections of these

two are therefore given by the equation

y^ {2am + am^) - iax {mx -y) = 0.

If these be at right angles, then

2am + am^ â€” 'iam = 0,

i.e. m= ^sJ2.

Ex. 2. From the point where any normal to the parabola y^ = ^ax

meets the axis is draion a line perpendicular to this normal ; prove that

this line always touches an equal parabola.

The equation of any normal to the parabola is

y = mx â€” 2am - am^.

This meets the axis in the point {2a + am'^, 0).

The equation to the straight line through this point perpendicular

to the normal is

y = wii {x-2a â€” am'^) ,

where m^m= - 1.

The equation is therefore

y = m,[x-2a-^^,

i.e. y = m^{x-2a) .

TANGENT AND NORMAL. EXAMPLES. 185

This straight line, as in Art. 206, always touches the equal parabola

y^= - 4a (a;- 2a),

whose vertex is the point (2a, 0) and whose concavity is towards the

negative end of the axis of x.

EXAMPLES. XXVI.

Write down the equations to the tangent and normal

1. at the point (4, 6) of the parabola y^=9x,

2. at the point of the parabola ?/^ = 6a; whose ordinate is 12,

3. at the ends of the latus rectum of the parabola y^ â€” 12x,

4. at the ends of the latus rectum of the parabola ^2 â€” 4.^ (.^ _ a).

5. Find the equation to that tangent to the parabola y^ = 7x

which is parallel to the straight line 4y -x + S = 0. Find also its

point of contact.

6. A tangent to the parabola y^=Aax makes an angle of 60Â° with

the axis ; find its point of contact.

7. A tangent to the parabola y'^ = 8x makes an angle of 45Â° with

the straight line y = Sx + 5. Find its equation and its point of

contact.

8. Find the points of the parabola y^ = 4:ax at which (i) the

tangent, and (ii) the normal is inclined at 30Â° to the axis.

9. Find the equation to the tangents to the parabola y^=9x which

goes through the point (4, 10).

10. Prove that the straight line x + y = l touches the parabola

y=x-x^.

11. Prove that the straight line y = mx + c touches the parabola

?/^=4a (a; + a) if c=ma + â€”.

' m

12. Prove that the straight line Ix + my + w = touches the parabola

y^=4:ax if ln=amP.

13. For what point of the parabola y^ = 4:ax is (1) the normal equal

to twice the subtangent, (2) the normal equal to the difference between

the subtangent and the subnormal ?

Find the equations to the common tangents of

14. the parabolas ?/2 = 4aa; and .'r2 = 4&i/,

15. the circle x^ + y^=4:ax and the parabola y^=4:ax.

16. Two equal parabolas have the same vertex and their axes are

at right angles ; prove that the common tangent touches each at the

end of a latus rectum.

186 COOKDINATE GEOMETRY. [ExS.

17. Prove that two tangents to the parabolas y^ â€” 4a {x + a) and

y^=4:a' {x + a'), which are at right angles to one another, meet on the

straight line x + a + a' = 0.

Shew also that this straight line is the common chord of the two

parabolas.

18. PN is an ordinate of the parabola ; a straight line is drawn

parallel to the axis to bisect NP and meets the curve in Q ; prove

that NQ meets the tangent at the vertex in a point T such that

AT = %NP.

19. Prove that the chord of the parabola y^ â€” 'iax, whose equation

isy -'XiJ2 + 4:a^2 = 0, is a normal to the curve and that its length is

6 ^Sa.

20. If perpendiculars be drawn on any tangent to a parabola from

two fixed points on the axis, which are equidistant from the focus,

prove that the difference of their squares is constant.

21. If P, Q, and R be three points on a parabola whose ordinates

are in geometrical progression, prove that the tangents at P and R

meet on the ordinate of Q.

22. Tangents are drawn to a parabola at points whose abscissae

are in the ratio fi : 1; prove that they intersect on the curve .

y^={fi^ + fi~^)^ax.

23. If the tangents at the points {x', y') and {x", y") meet at the

point [x-^, y-j) and the normals at the same points in {x^, y^, prove

that

(1) .,=y^ .ni y,=y^f ,

(2) .,=2a + ^'^Â±^-;^^ and V.^-yV^^,

and hence that

(3) x,=2a+y-^- X, and y, = - "^^^^ .

24. From the preceding question prove that, if tangents be drawn

to the parabola y^ = 4:ax from any point on the parabola y^ â€” a{x+h),

then the normals at the points of contact meet on a fixed straight

line.

25. Find the lengths of the normals drawn from the point on the

axis of the parabola y^ = 8ax whose distance from the focus is 8a.

26. Prove that the locus of the middle point of the portion of a

normal intersected between the curve and the axis is a parabola whose

vertex is the focus and whose latus rectum is one quarter of that of

the original parabola.

27. Prove that the distance between a tangent to the parabola and

the parallel normal is a cosec 6 sec^ 6, where 6 is the angle that either

makes with the axis.

XXVI.l TANGENT AND NORMAL. EXAMPLES. 187

28. PNP' is a double ordinate of the parabola ; prove that the

locus of the point of intersection of the normal at P and the diameter

through P' is the equal parabola y^ = 4a (x-Aa).

29. The normal at any point P meets the axis in G and the

tangent at the vertex in G' ; HA be the vertex and the rectangle

AGQG' he completed, prove that the equation to the locus of Q is

30. Two equal parabolas have the same focus and their axes are

at right angles ; a normal to one is perpendicular to a normal to the

other ; prove that the locus of the point of intersection of these

normals is another parabola.

31. If a normal to a parabola make an angle with the axis,

shew that it will cut the curve again at an angle tan~i (^ tan 0).

32. Prove that the two parabolas y^ = 4:ax and y^=4:c{x- b) cannot

have a common normal, other than the axis, unless >2.

a-c

33. If aP>8h-, prove that a point can be found such that the two

tangents from it to the parabola y^=4tax are normals to the parabola

x^=^by.

34. Prove that three tangents to a parabola, which are such that

the tangents of their inclinations to the axis are in a given harmonical

progression, form a triangle whose area is constant.

35. Prove that the parabolas y^=4tax and x^ = 4:by cut one another

at an angle tan ^

2 {aÂ« + 6Â«)

36. Prove that two parabolas, having the same focus and their axes

in opposite directions, cut at right angles.

37. Shew that the two parabolas

x^ + 4:a{y-2b-a) = and y'^ = 4:b{x-2a + b)

intersect at right angles at a common end of the latus rectum

of each.

38. ^ parabola is drawn touching the axis of x at the origin and

having its vertex at a given distance k from this axis. Prove that the

axis of the parabola is a tangent to the parabola x'^= -Sk {y -2k).

211. Some properties of the Parabola.

(a) If the tangent and normal at any point P of the

parabola meet the axis in T and G respectively, then

188

COORDINATE GEOMETRY.

and the tangent at P is equally inclined to the axis and the

focal distance of P.

Let P be the point (x, y).

Draw PM perpendicular to the directrix.

By Art. 209, we have AT^AN.

:. TS=TA + AS=^AF+ZA = ZF=MP = SP,

and hence z STP = z SPT.

By the same article, NG - "iAS = ZS.

:. SG^SN-\-NG = ZS+SF=MP = SP.

(/8) If the tangent at P meet the directrix in K, then

KSP is a right angle.

Por z SPT=^ L PTS=^ L KPM.

Hence the two triangles KPS and KPM have the two

sides KPj PS and the angle KPS equal respectively to the

two sides KP, PM and the angle KPM.

Hence z KSP = z KMP = a right angle.

Also lSKP=lMKP.

(y) Tangents at the extremities of any focal chord inter-

sect at right angles in the directrix.

For, if PS be produced to meet the curve in P', then,

since z P'SK is a right angle, the tangent at P' meets the

directrix in K

PROPERTIES OF THE PARABOLA. 189

Also, by (13), L MKP = z SEP,

and, similarly, / M'KP' - L SKF.

Hence

z PKP' = J z: SKM + 1 z SKM' = a right angle.

(8) 7/ /Sl^ 6e jyerpe^Lclicular to the tangent at P, then Y

lies on the tangent at the vertex and SY^ = AS . SP.

For the equation to any tangent is

yâ€”mx-\ â€” (Ij.

The equation to the perpendicular to, this line passing

through the focus is

2/ = - (^-Â«) (2).

The lines (1) and (2) meet where

a \ , . 1 a

nix H â€” =â€” â€”[xâ€” a) = X -^ â€” 5

7n m m in

i. e. where x â€” 0.

Hence Y lies on the tangent at the vertex.

Also, by Euc. vi. 8, Cor.,

SY^ = SA.ST=AS.SP,

212. To prove that through any given point {x^^ y^

there pass, in general, two tangents to the parabola.

The equation to any tangent is (by Art. 206)

y = mx -\ â€” ( 1 ).

If this pass through the fixed point (x^, y^), we have

a

y, = TUX, + â€” ,

i. e. m^Xj^ â€” tny^ + ^ = (2).

For any given values of x^ and y^ this equation is in

general a quadratic equation and gives two values of m

(real or imaginary).

Corresponding to each value of in we have, by substi-

tuting in (1), a different tangent.

190 COORDINATE GEOMETRY.

The roots of (2) are real and different if y-^ â€” 4:ax-^ be

positive, i.e., by Art. 201, if the point {x-^, y-^) lie without

the curve.

They are equal, i. e. the two tangents coalesce into one

tangent, if yiâ€” ^cix^ be zero, i.e. if the point {x-^, y^ lie on

the curve.

The two roots are imaginary if y^ â€” 4a.x\ be negative,

i.e. if the point (cCj, y^ lie within the curve.

213. Equation to the chord of contact of tangents

drawn from a point {x^, y^).

The equation to the tangent at any point Q, whose

coordinates are x' and y', is

yy' = 2a (x + x).

Also the tangent at the point E, whose coordinates are

x" and y", is

yy" â€” 2a{x + x").

If these tangents meet at the point T, whose coordi-

nates are x^ and y^, we have

y^y' = 2a{x^+ x) (1)

and y^y" = 2a{x^ + x") (2).

The equation to QR is then

3ryi = 2a(x + Xi) (3).

For, since (1) is true, the point {x, y') lies on (3).

Also, since (2) is true, the point {x", y") lies on (3).

Hence (3) must be the equation to the straight line

joining ix\ y) to the point {x' , y"), i. e. it must be the

equation to QR the chord of contact of tangents from the

point {x^, ?/i).

214. The polar of any point with respect to a para-

bola is defined as in Art. 162.

To find the equation of the polar of the point [x^ , 2/1)

with respect to the parabola y^ â€” ^ax.

Let Q and R be the points in which any chord drawn

through the point P, whose coordinates are (x^, y^), meets

the parabola.

THE PARABOLA. POLE AND POLAR.

191

Let the tangents at Q and R meet in the point whose

coordinates are (A, k).

T(h.Wji.

We require the locus of (h, k).

Since ^^ is the chord of contact of tangents from (7i, k)

its equation (Art. 213) is

ky = 2a(x + h).

Since this straight line passes through the point (r^ , y^)

we haye

%i = 2a{x^ + h) (1).

Since the relation (1) is true, it follows that the point

{hj k) always lies on the straight line

3ryi = 2a(x + xJ (2).

Hence (2) is the equation to the polar of (ic^, y^.

Cor. The equation to the polar of the focus, viz. the point [a, 0),

is Q = x + a, so that the polar of the focus is the directrix.

215. When the point (x-^,y^ lies without the parabola

the equation to its polar is the same as the equation to the

chord of contact of tangents drawn from [x-^^, y^).

When (x^, y^) is on the parabola the polar is the same

as the tangent at the point.

As in Art. 164 the polar of (a^, y^) might have been

defined as the chord of contact of the tangents (real or

imaginary) that can be drawn from it to the parabola.

216. Geometrical construction for the polar of a point

192

COORDINATE GEOMETRY.

Let T be the point {x^^ 2/1)3 so that its polar is

yy^=-2a{x + x^) (1).

Through T draw a straight line parallel to the axis ; its

equation is therefore

y=yi (2).

Let this straight line meet the polar

in V and the curve in P.

The coordinates of F, which is the

intersection of (1) and (2), are therefore

^ â€”x^ and 2/1 (3).

Also P is the point on the curve

whose ordinate is y^, and whose coordi-

nates are therefore

2

and 2/1.

yi

4:a

Since abscissa of P=

abscissa of :Z^ + abscissa of V

there-

fore, by Art. 22, Cor., P

middle point of TV.

Also the tangent at P is

2/1'

is

the

yy,= 2a^.^f^

which is parallel to (1).

Hence the polar of T is parallel

to the tangent at P.

To draw the polar of T we therefore draw a line through

T, parallel to the axis, to meet the curve in P and produce

it to Fso that TP-PV; a line through F parallel to the

tangent at P is then the polar required.

217. If the polar of a point P passes through the point T, then

the polar of T goes through P. (Fig. Art. 214).

Let P be the point (x^, y-^) and T the point {h, k).

The polar of P is yy^ = 2a{x + x^).

Since it passes through T, we have

yj^k = 2a{x-^ + h) (1).

PAIR OF TANGENTS FROM ANY POINT. 193

The polar of T isyk = 2a (x+h).

Since (1) is true, this equatio n is satisfied by the coordinates Xj^

and t/i-

Hence the proposition.

Cor. The point of intersection, T, of the polar s of two points,

P and Q, is the pole of the line PQ.

218. To find the pole of a given straight line ivith respect to the

parabola.

Let the given straight line be

Ax + By+C=0.

If its pole be the point {x^, y-^), it must be the same straight

line as

yy^ = 2a{x + x^),

i.e. 2ax - yyi + 2axj^ = 0.

Since these straight lines are the same, we have

2a _ -yi _ 2axi

G ^ 2Ba

I.e. xi = j and y^= - -j- -

219. To find the equation to the pcdr of tangents that

can he drawn to the parabola from the point {x^^ y^.

Let (A, k) be any point on either of the tangents drawn

from (rL'i, y^. The equation to the line joining (x^, y^) to

(^, k) is

kâ€” y. hy, - kx.

%.e. y = - â€” -x^-^ \

If this be a tangent it must be of the form

a

y â€” mx -{ â€” ,

, , . k â€” y^ , hy. â€” kx. a

so that . â€” ^ = m and ~- i = â€” .

a â€” x^ h â€” x^ m

Hence, by multiplication,

k â€” y^ hy^ â€” kx^

i. e. a (lb - x^^ = {k â€” y^) [hy^ â€” kx^.

I^ 13

194 COOKDINATE GEOMETRY.

The locus of the point (A, k) {i. e. the pair of tangents

required) is therefore

a(x-x^y = {y-y^) {xy^-yx:^ (1).

It will be seen that this equation is the same as

{f - \ax) (2/1^ - 4arci) = {2/2/1 - 2Â« (a? + x^f.

220. To prove that the middle points of a system of

parallel chords of a parabola all lie on a straight line which

is parallel to the axis.

Since the chords are all parallel, they all make the same

angle with the axis of x. Let Q

the tangent of this angle be on.

The equation to QB, any-

one of these chords, is there- y^^-

fore

y - mx + c (1 ), '^,

where c is different for the

several chords, but 7n is the

same.

This straight line meets the parabola y^ = 4:ax in points

whose ordinates are given by

m,y^ = 4:a (y â€” c),

4:a Aac , .

I.e. V y + =^0 (2).

^ m ^ m â€¢ '

Let the roots of this equation, i.e. the ordinates of Q

and Rj be y' and y'\ and let the coordinates of F, the

middle point of QR, be (h, k).

Then, by Art. 22,

T _ y + y" _ 2Â«

2 m

from equation (2).

The coordinates of V therefore satisfy the equation

2a

y=m^

so that the locus of F is a straight line parallel to the axis

of the curve,

MIDDLE POINTS OF PARALLEL CHORDS. 195

2a

The straight line 3/ = â€” meets the curve in a point P,

whose ordinate is â€” and whose abscissa is therefore â€” x .

m m"

The tangent at this point is, by Art. 205,

a

y = Tnx -\ â€” ,

and is therefore parallel to each of the given chords.

Hence the locus of the middle points of a system of

parallel chords of a parabola is a straight line which is

parallel to the axis and meets the curve at a point the

tangent at which is parallel to the given system.

221. To find the equation to the chord of the parabola ivhich is

bisected at any point {h, Jc).

By the last article the required chord is parallel to the tangent at

the point P where a line through {h, k) parallel to the axis meets the

curve.

Also, by Art. 216, the polar of {h, k) is parallel to the tangent at

this same point P.

The required chord is therefore parallel to the polar yJc = 2a {x + h).

Hence, since it goes through {h, k), its equation is

k{y-k) = 2a{x- h) (Art. 67).

222. Diameter. Def. The locus of the middle points

of a system of parallel chords of a parabola is called a

diameter and the chords are called its ordinates.

Thus, in the figure of Art. 220, PF is a diameter and

QB and all the parallel chords are ordinates to this

diameter.

The proposition of that article may therefore be stated

as follows.

Any diameter of a parabola is parallel to the axis and

the tangent at the point where it Tneets the curve is parallel

to its ordinates.

223. The tangents at the ends of any chord meet on

the diameter which bisects the chord.

Let the equation of QR (Fig., Art. 220) be

y = mx + c (1),

13â€”2

196

COORDINATE GEOMETRY.

and let the tangents at Q and R meet at the point T

Then QR is the chord of contact of tangents drawn

from T^ and hence its equation is

2/2/1 = 2a{x + x^) (Art. 213).

Comparing this with equation (1), we have

2a ,, , 2a

â€” = m, so that Vi = â€” ->

2/1 ^*

and therefore T lies on the straight line

2a

^ m

But this straight line was proved, in Art. 220, to be

the diameter P V which bisects the chord.

224. To find the equation to a parabola, the axes

being any diameter and the tangent to the parabola at the

point where this diameter meets the curve.

Let PVX be the diameter and PY the tangent at P

meeting the axis in T.

Take any point Q on the curve,

and draw QM perpendicular to the

axis meeting the diameter P F in L.

Let PVhQ X and VQ be y.

Draw PN perpendicular to the

axis of the curve, and let

e^ /. YPX=iPTM,

Then

iAS. A]S[^PN^ = ]SfT^ ts,Ti^e=^.AN^ . tan^ 6.

:. ANr=:AS. cot^ e = a cot^ e,

and PN = JIASTaN = 2a cot 6.

Now QM'- = 4:AS.AM=4:a.AM (1).

Also

QM=JSrP + LQ = 2acote+ VQsmO = 2acotO+ysinO,

and AM=A]\/' + PV+ VL=-acot^e + x + ycose.

THE PARABOLA. EXAMPLES. 197

Substituting these values in (1), we have

(2a cot + y sin Oy â€” ia (a cot^ + x + y cos 6),

i. e. if- sin^ 6 â€” ^ax.

The required equation is therefore

y'^^lpx (2),

where

p - T^= Â« (1 + ^ot' Q) = a^ AN= SP (by Art. 202).

The equation to the parabola referred to the above axes

is therefore of the same form as its equation referred to the

rectangular axes of Art. 197.

The equation (2) states that

QV'^^iSP.PV.

225. The quantity 4^j is called the parameter of the

diameter P V. It is equal in length to the chord which is

parallel to P F and passes through the focus.

For if Q'V'R' be the chord, parallel to PZand passing

through the focus and meeting PT in V\ we have

PY' = ST=SP^p,

so that Q' V"" ^ip.PV'^ ip\

and hence Q'R' =-'2Q'V' ^ ip.

226. Just as in Art. 205 it could now be shown that

the tangent at any point {x\ y) of the above curve is

yy â€” 2p (x + x).

Similarly for the equation to the polar of any point.

EXAMPLES. XXVII.

1. Prove that the length of the chord joining the points of

contact of tangents drawn from the point (Xj, y^ is

ijy-^ + 4a2 fjy^^ - 4aa; J ^

a

2. Prove that the area of the triangle formed by the tangents

3

from the point {x^^ y^ and the chord of contact is {y^ - ^ax^^ -^2a.

198 COORDINATE GEOMETRY. [Exs. XXVII.]

3. If a perpendicular be let fall from any point P upon its polar

prove that the distance of the foot of this perpendicular from the

focus is equal to the distance of the point P from the directrix.

4. What is the equation to the chord of the parabola y^ = 8x

which is bisected at the point (2, - 3) ?

5. The general equation to a system of parallel chords in the

parabola y^ = ^x is 4:X-y + k = 0.

Wliat is the equation to the corresponding diameter ?

6. P, Q, and B are three points on a parabola and the chord PQ

cuts the diameter through R in V. Ordinates P3I and QN are drawn

to this diameter. Prove that RM . RN=RV^.

7. Two equal parabolas with axes in opposite directions touch at

a point O. From a point P on one of them are drawn tangents PQ

and PQ' to the other. Prove that QQ' will touch the first parabola in

â– P' where PP' is parallel to the common tangent at O.

Coordinates of any point on the parabola ex-

and y"^^4.aa/' (3).

TANGENT AT ANY POINT OF A PARABOLA. 181

Hence, by subtraction, we have

2/"' - y"' ^ 4a (x" - x),

i.e.

{y"-y'){y" + y') = 4.a{x"-x'),

and hence

y" -y 4o&

X â€”X y -^ y

Substituting this value in equation (1), we have, as

the equation to any secant PQ^

4a

i.e. y {y + y") = ^ax + y'y" + y^ â€” 4:ax

â€” iax + y'y" (4).

To obtain the equation of the tangent at (x\ y') we take

Q indefinitely close to P, and hence, in the limit, put y" = y .

The equation (4) then becomes

^yy â€” y"^ + 4aa; = 4^ax + 4aa;',

i.e. yy' = 2a(x + x').

Cor. It will be noted that the equation to the tangent

is obtained from the equation to the curve by the rule of

Art. 152.

Exs. The equation to the tangent at the point {2, -4) of the

parabola y^=Qx is

2/(-4) = 4(a; + 2),

i.e. x-\-y + 1 = 0.

The equation to the tangent at the point ( â€” 2 Â» â€” ) of the parabola

y^=4iax is

2a . f a\

y . â€” = 2a[x + ~] ,

d

I.e. ii = mx-\ â€” .

m

206. To find the condition that the straight line

y = mx + c (1)

rtiay touch the parabola y^ â€” 4:ax (2).

The abscisses of the points in which the straight line (1)

meets the curve (2) are as in Art. 203, given by the equation

rn^x^ + 2x {mo - 2a) + c^ = (3).

182 COORDINATE GEOMETRY.

The line (1) will touch (2) if it meet it in two points

which are indefinitely close to one another, i.e. in two

points which ultimately coincide.

The roots of equation (3) must therefore be equal.

The condition for this is

4 {mc â€” 2ay â€” iirrc^

i.e. ct? - amc â€” 0,

a

so that G = â€” .

m

Substituting this value of c in (1), we have as the

equation to a tangent,

a

y = mx + â€” .

m

In this equation tyi is the tangent of th6 angle which

the tangent makes with the axis of x.

The foregoing proposition may also be obtained from the equation

of Art. 205.

For equation (4) of that article may be written

la lax' ,^.

y=â€”x+^ (1).

y y

In this equation put â€” , =m, i.e. y =â€” ,

^ y m

, y'^ a , 2ax' a

and hence x = -r- = ^, , and â€” â€” = â€” .

4a m^ y m

a

The equation (1) then becomes y = mx-] â€” .

Also it is the tangent at the point {x', y'), i.e. (â€”^, â€” j .

207. Equation to the normal at (x, y'). The required

normal is the straight line which passes through the point

{x\ y') and is perpendicular to the tangent, i.e. to the

straight line

2a . ,.

yz=zâ€”-{x + x).

Its equation is therefore

y â€” y = rri' {x â€” x'),

2fi 11

where m' x â€” rr - 1 i.e. ^i' = -^ (Art. 69.)

y 2a

NORMAL TO A PARABOLA.

183

,(1).

and the equation to the normal is

y-y'=^(x-x')

208. To exj^ress the equation of the normal in the form

y = mx â€” 2a7n â€” am^.

In equation (1) of the last article put

2a

= m, %.e. y

- â€” ^.

am.

/2

Hence

, y 2

X =^ = aiir.

4a

The normal is therefore

y + 2am = m {x

am

%

t.e.

y = mx â€” 2am â€” am^

and it is a normal at the point (am^, â€” 2am) of the curve.

In this equation m is the tangent of the angle which

the normal makes with the axis. It must be carefully

distinguished from the m of Art. 206 which is the tangent

of the angle which the tangent makes with the axis. The

" m" of this article is - 1 divided by the " m'' of Art. 206.

209. Subtangent and Subnormal. Def. If

the tangent and normal at any point P of a conic section

meet the axis in T and G respectively and PN be the

ordinate at P, then NT is called the Subtangent and NG the

Subnormal of P.

To find the length of the subtangent and suhnorm^al.

If P be the point {x\ y') the equation to TP is, by

Art. 205,

yy â€”2a{x-\- x) (1).

To obtain the length of AT^ we

have to find the point where this

straight line meets the axis of tc,

i.e. we put 2/ = 0in (1) and we

have

x=^-x (2).

Hence AT=AN,

184 COORDINATE GEOMETRY.

[The negative sign in equation (2) shews that T and

N always lie on opposite sides of the vertex -4.]

Hence the subtangent iV^= 2J^iV = twice the abscissa

of the point P.

Since TFG is a right-angled triangle, we have (Euc. vi. 8)

FN'^^TN.NG.

Hence the subnormal NG

_ PiP _ PN^

The subnormal is therefore constant for all points on

the parabola and is equal to the semi-latus rectum.

210. Ex. 1. If a chord which is normal to the parabola at one

end subtend a right angle at the vertex, prove that it is inclined at an

angle tan~^ ^J2 to the axis.

The equation to any chord which is normal is

y = mx â€” 2am - am^,

i.e. mx-y = 2am+am^.

The parabola is y^ â€” 4:ax.

The straight lines joining the origin to the intersections of these

two are therefore given by the equation

y^ {2am + am^) - iax {mx -y) = 0.

If these be at right angles, then

2am + am^ â€” 'iam = 0,

i.e. m= ^sJ2.

Ex. 2. From the point where any normal to the parabola y^ = ^ax

meets the axis is draion a line perpendicular to this normal ; prove that

this line always touches an equal parabola.

The equation of any normal to the parabola is

y = mx â€” 2am - am^.

This meets the axis in the point {2a + am'^, 0).

The equation to the straight line through this point perpendicular

to the normal is

y = wii {x-2a â€” am'^) ,

where m^m= - 1.

The equation is therefore

y = m,[x-2a-^^,

i.e. y = m^{x-2a) .

TANGENT AND NORMAL. EXAMPLES. 185

This straight line, as in Art. 206, always touches the equal parabola

y^= - 4a (a;- 2a),

whose vertex is the point (2a, 0) and whose concavity is towards the

negative end of the axis of x.

EXAMPLES. XXVI.

Write down the equations to the tangent and normal

1. at the point (4, 6) of the parabola y^=9x,

2. at the point of the parabola ?/^ = 6a; whose ordinate is 12,

3. at the ends of the latus rectum of the parabola y^ â€” 12x,

4. at the ends of the latus rectum of the parabola ^2 â€” 4.^ (.^ _ a).

5. Find the equation to that tangent to the parabola y^ = 7x

which is parallel to the straight line 4y -x + S = 0. Find also its

point of contact.

6. A tangent to the parabola y^=Aax makes an angle of 60Â° with

the axis ; find its point of contact.

7. A tangent to the parabola y'^ = 8x makes an angle of 45Â° with

the straight line y = Sx + 5. Find its equation and its point of

contact.

8. Find the points of the parabola y^ = 4:ax at which (i) the

tangent, and (ii) the normal is inclined at 30Â° to the axis.

9. Find the equation to the tangents to the parabola y^=9x which

goes through the point (4, 10).

10. Prove that the straight line x + y = l touches the parabola

y=x-x^.

11. Prove that the straight line y = mx + c touches the parabola

?/^=4a (a; + a) if c=ma + â€”.

' m

12. Prove that the straight line Ix + my + w = touches the parabola

y^=4:ax if ln=amP.

13. For what point of the parabola y^ = 4:ax is (1) the normal equal

to twice the subtangent, (2) the normal equal to the difference between

the subtangent and the subnormal ?

Find the equations to the common tangents of

14. the parabolas ?/2 = 4aa; and .'r2 = 4&i/,

15. the circle x^ + y^=4:ax and the parabola y^=4:ax.

16. Two equal parabolas have the same vertex and their axes are

at right angles ; prove that the common tangent touches each at the

end of a latus rectum.

186 COOKDINATE GEOMETRY. [ExS.

17. Prove that two tangents to the parabolas y^ â€” 4a {x + a) and

y^=4:a' {x + a'), which are at right angles to one another, meet on the

straight line x + a + a' = 0.

Shew also that this straight line is the common chord of the two

parabolas.

18. PN is an ordinate of the parabola ; a straight line is drawn

parallel to the axis to bisect NP and meets the curve in Q ; prove

that NQ meets the tangent at the vertex in a point T such that

AT = %NP.

19. Prove that the chord of the parabola y^ â€” 'iax, whose equation

isy -'XiJ2 + 4:a^2 = 0, is a normal to the curve and that its length is

6 ^Sa.

20. If perpendiculars be drawn on any tangent to a parabola from

two fixed points on the axis, which are equidistant from the focus,

prove that the difference of their squares is constant.

21. If P, Q, and R be three points on a parabola whose ordinates

are in geometrical progression, prove that the tangents at P and R

meet on the ordinate of Q.

22. Tangents are drawn to a parabola at points whose abscissae

are in the ratio fi : 1; prove that they intersect on the curve .

y^={fi^ + fi~^)^ax.

23. If the tangents at the points {x', y') and {x", y") meet at the

point [x-^, y-j) and the normals at the same points in {x^, y^, prove

that

(1) .,=y^ .ni y,=y^f ,

(2) .,=2a + ^'^Â±^-;^^ and V.^-yV^^,

and hence that

(3) x,=2a+y-^- X, and y, = - "^^^^ .

24. From the preceding question prove that, if tangents be drawn

to the parabola y^ = 4:ax from any point on the parabola y^ â€” a{x+h),

then the normals at the points of contact meet on a fixed straight

line.

25. Find the lengths of the normals drawn from the point on the

axis of the parabola y^ = 8ax whose distance from the focus is 8a.

26. Prove that the locus of the middle point of the portion of a

normal intersected between the curve and the axis is a parabola whose

vertex is the focus and whose latus rectum is one quarter of that of

the original parabola.

27. Prove that the distance between a tangent to the parabola and

the parallel normal is a cosec 6 sec^ 6, where 6 is the angle that either

makes with the axis.

XXVI.l TANGENT AND NORMAL. EXAMPLES. 187

28. PNP' is a double ordinate of the parabola ; prove that the

locus of the point of intersection of the normal at P and the diameter

through P' is the equal parabola y^ = 4a (x-Aa).

29. The normal at any point P meets the axis in G and the

tangent at the vertex in G' ; HA be the vertex and the rectangle

AGQG' he completed, prove that the equation to the locus of Q is

30. Two equal parabolas have the same focus and their axes are

at right angles ; a normal to one is perpendicular to a normal to the

other ; prove that the locus of the point of intersection of these

normals is another parabola.

31. If a normal to a parabola make an angle with the axis,

shew that it will cut the curve again at an angle tan~i (^ tan 0).

32. Prove that the two parabolas y^ = 4:ax and y^=4:c{x- b) cannot

have a common normal, other than the axis, unless >2.

a-c

33. If aP>8h-, prove that a point can be found such that the two

tangents from it to the parabola y^=4tax are normals to the parabola

x^=^by.

34. Prove that three tangents to a parabola, which are such that

the tangents of their inclinations to the axis are in a given harmonical

progression, form a triangle whose area is constant.

35. Prove that the parabolas y^=4tax and x^ = 4:by cut one another

at an angle tan ^

2 {aÂ« + 6Â«)

36. Prove that two parabolas, having the same focus and their axes

in opposite directions, cut at right angles.

37. Shew that the two parabolas

x^ + 4:a{y-2b-a) = and y'^ = 4:b{x-2a + b)

intersect at right angles at a common end of the latus rectum

of each.

38. ^ parabola is drawn touching the axis of x at the origin and

having its vertex at a given distance k from this axis. Prove that the

axis of the parabola is a tangent to the parabola x'^= -Sk {y -2k).

211. Some properties of the Parabola.

(a) If the tangent and normal at any point P of the

parabola meet the axis in T and G respectively, then

188

COORDINATE GEOMETRY.

and the tangent at P is equally inclined to the axis and the

focal distance of P.

Let P be the point (x, y).

Draw PM perpendicular to the directrix.

By Art. 209, we have AT^AN.

:. TS=TA + AS=^AF+ZA = ZF=MP = SP,

and hence z STP = z SPT.

By the same article, NG - "iAS = ZS.

:. SG^SN-\-NG = ZS+SF=MP = SP.

(/8) If the tangent at P meet the directrix in K, then

KSP is a right angle.

Por z SPT=^ L PTS=^ L KPM.

Hence the two triangles KPS and KPM have the two

sides KPj PS and the angle KPS equal respectively to the

two sides KP, PM and the angle KPM.

Hence z KSP = z KMP = a right angle.

Also lSKP=lMKP.

(y) Tangents at the extremities of any focal chord inter-

sect at right angles in the directrix.

For, if PS be produced to meet the curve in P', then,

since z P'SK is a right angle, the tangent at P' meets the

directrix in K

PROPERTIES OF THE PARABOLA. 189

Also, by (13), L MKP = z SEP,

and, similarly, / M'KP' - L SKF.

Hence

z PKP' = J z: SKM + 1 z SKM' = a right angle.

(8) 7/ /Sl^ 6e jyerpe^Lclicular to the tangent at P, then Y

lies on the tangent at the vertex and SY^ = AS . SP.

For the equation to any tangent is

yâ€”mx-\ â€” (Ij.

The equation to the perpendicular to, this line passing

through the focus is

2/ = - (^-Â«) (2).

The lines (1) and (2) meet where

a \ , . 1 a

nix H â€” =â€” â€”[xâ€” a) = X -^ â€” 5

7n m m in

i. e. where x â€” 0.

Hence Y lies on the tangent at the vertex.

Also, by Euc. vi. 8, Cor.,

SY^ = SA.ST=AS.SP,

212. To prove that through any given point {x^^ y^

there pass, in general, two tangents to the parabola.

The equation to any tangent is (by Art. 206)

y = mx -\ â€” ( 1 ).

If this pass through the fixed point (x^, y^), we have

a

y, = TUX, + â€” ,

i. e. m^Xj^ â€” tny^ + ^ = (2).

For any given values of x^ and y^ this equation is in

general a quadratic equation and gives two values of m

(real or imaginary).

Corresponding to each value of in we have, by substi-

tuting in (1), a different tangent.

190 COORDINATE GEOMETRY.

The roots of (2) are real and different if y-^ â€” 4:ax-^ be

positive, i.e., by Art. 201, if the point {x-^, y-^) lie without

the curve.

They are equal, i. e. the two tangents coalesce into one

tangent, if yiâ€” ^cix^ be zero, i.e. if the point {x-^, y^ lie on

the curve.

The two roots are imaginary if y^ â€” 4a.x\ be negative,

i.e. if the point (cCj, y^ lie within the curve.

213. Equation to the chord of contact of tangents

drawn from a point {x^, y^).

The equation to the tangent at any point Q, whose

coordinates are x' and y', is

yy' = 2a (x + x).

Also the tangent at the point E, whose coordinates are

x" and y", is

yy" â€” 2a{x + x").

If these tangents meet at the point T, whose coordi-

nates are x^ and y^, we have

y^y' = 2a{x^+ x) (1)

and y^y" = 2a{x^ + x") (2).

The equation to QR is then

3ryi = 2a(x + Xi) (3).

For, since (1) is true, the point {x, y') lies on (3).

Also, since (2) is true, the point {x", y") lies on (3).

Hence (3) must be the equation to the straight line

joining ix\ y) to the point {x' , y"), i. e. it must be the

equation to QR the chord of contact of tangents from the

point {x^, ?/i).

214. The polar of any point with respect to a para-

bola is defined as in Art. 162.

To find the equation of the polar of the point [x^ , 2/1)

with respect to the parabola y^ â€” ^ax.

Let Q and R be the points in which any chord drawn

through the point P, whose coordinates are (x^, y^), meets

the parabola.

THE PARABOLA. POLE AND POLAR.

191

Let the tangents at Q and R meet in the point whose

coordinates are (A, k).

T(h.Wji.

We require the locus of (h, k).

Since ^^ is the chord of contact of tangents from (7i, k)

its equation (Art. 213) is

ky = 2a(x + h).

Since this straight line passes through the point (r^ , y^)

we haye

%i = 2a{x^ + h) (1).

Since the relation (1) is true, it follows that the point

{hj k) always lies on the straight line

3ryi = 2a(x + xJ (2).

Hence (2) is the equation to the polar of (ic^, y^.

Cor. The equation to the polar of the focus, viz. the point [a, 0),

is Q = x + a, so that the polar of the focus is the directrix.

215. When the point (x-^,y^ lies without the parabola

the equation to its polar is the same as the equation to the

chord of contact of tangents drawn from [x-^^, y^).

When (x^, y^) is on the parabola the polar is the same

as the tangent at the point.

As in Art. 164 the polar of (a^, y^) might have been

defined as the chord of contact of the tangents (real or

imaginary) that can be drawn from it to the parabola.

216. Geometrical construction for the polar of a point

192

COORDINATE GEOMETRY.

Let T be the point {x^^ 2/1)3 so that its polar is

yy^=-2a{x + x^) (1).

Through T draw a straight line parallel to the axis ; its

equation is therefore

y=yi (2).

Let this straight line meet the polar

in V and the curve in P.

The coordinates of F, which is the

intersection of (1) and (2), are therefore

^ â€”x^ and 2/1 (3).

Also P is the point on the curve

whose ordinate is y^, and whose coordi-

nates are therefore

2

and 2/1.

yi

4:a

Since abscissa of P=

abscissa of :Z^ + abscissa of V

there-

fore, by Art. 22, Cor., P

middle point of TV.

Also the tangent at P is

2/1'

is

the

yy,= 2a^.^f^

which is parallel to (1).

Hence the polar of T is parallel

to the tangent at P.

To draw the polar of T we therefore draw a line through

T, parallel to the axis, to meet the curve in P and produce

it to Fso that TP-PV; a line through F parallel to the

tangent at P is then the polar required.

217. If the polar of a point P passes through the point T, then

the polar of T goes through P. (Fig. Art. 214).

Let P be the point (x^, y-^) and T the point {h, k).

The polar of P is yy^ = 2a{x + x^).

Since it passes through T, we have

yj^k = 2a{x-^ + h) (1).

PAIR OF TANGENTS FROM ANY POINT. 193

The polar of T isyk = 2a (x+h).

Since (1) is true, this equatio n is satisfied by the coordinates Xj^

and t/i-

Hence the proposition.

Cor. The point of intersection, T, of the polar s of two points,

P and Q, is the pole of the line PQ.

218. To find the pole of a given straight line ivith respect to the

parabola.

Let the given straight line be

Ax + By+C=0.

If its pole be the point {x^, y-^), it must be the same straight

line as

yy^ = 2a{x + x^),

i.e. 2ax - yyi + 2axj^ = 0.

Since these straight lines are the same, we have

2a _ -yi _ 2axi

G ^ 2Ba

I.e. xi = j and y^= - -j- -

219. To find the equation to the pcdr of tangents that

can he drawn to the parabola from the point {x^^ y^.

Let (A, k) be any point on either of the tangents drawn

from (rL'i, y^. The equation to the line joining (x^, y^) to

(^, k) is

kâ€” y. hy, - kx.

%.e. y = - â€” -x^-^ \

If this be a tangent it must be of the form

a

y â€” mx -{ â€” ,

, , . k â€” y^ , hy. â€” kx. a

so that . â€” ^ = m and ~- i = â€” .

a â€” x^ h â€” x^ m

Hence, by multiplication,

k â€” y^ hy^ â€” kx^

i. e. a (lb - x^^ = {k â€” y^) [hy^ â€” kx^.

I^ 13

194 COOKDINATE GEOMETRY.

The locus of the point (A, k) {i. e. the pair of tangents

required) is therefore

a(x-x^y = {y-y^) {xy^-yx:^ (1).

It will be seen that this equation is the same as

{f - \ax) (2/1^ - 4arci) = {2/2/1 - 2Â« (a? + x^f.

220. To prove that the middle points of a system of

parallel chords of a parabola all lie on a straight line which

is parallel to the axis.

Since the chords are all parallel, they all make the same

angle with the axis of x. Let Q

the tangent of this angle be on.

The equation to QB, any-

one of these chords, is there- y^^-

fore

y - mx + c (1 ), '^,

where c is different for the

several chords, but 7n is the

same.

This straight line meets the parabola y^ = 4:ax in points

whose ordinates are given by

m,y^ = 4:a (y â€” c),

4:a Aac , .

I.e. V y + =^0 (2).

^ m ^ m â€¢ '

Let the roots of this equation, i.e. the ordinates of Q

and Rj be y' and y'\ and let the coordinates of F, the

middle point of QR, be (h, k).

Then, by Art. 22,

T _ y + y" _ 2Â«

2 m

from equation (2).

The coordinates of V therefore satisfy the equation

2a

y=m^

so that the locus of F is a straight line parallel to the axis

of the curve,

MIDDLE POINTS OF PARALLEL CHORDS. 195

2a

The straight line 3/ = â€” meets the curve in a point P,

whose ordinate is â€” and whose abscissa is therefore â€” x .

m m"

The tangent at this point is, by Art. 205,

a

y = Tnx -\ â€” ,

and is therefore parallel to each of the given chords.

Hence the locus of the middle points of a system of

parallel chords of a parabola is a straight line which is

parallel to the axis and meets the curve at a point the

tangent at which is parallel to the given system.

221. To find the equation to the chord of the parabola ivhich is

bisected at any point {h, Jc).

By the last article the required chord is parallel to the tangent at

the point P where a line through {h, k) parallel to the axis meets the

curve.

Also, by Art. 216, the polar of {h, k) is parallel to the tangent at

this same point P.

The required chord is therefore parallel to the polar yJc = 2a {x + h).

Hence, since it goes through {h, k), its equation is

k{y-k) = 2a{x- h) (Art. 67).

222. Diameter. Def. The locus of the middle points

of a system of parallel chords of a parabola is called a

diameter and the chords are called its ordinates.

Thus, in the figure of Art. 220, PF is a diameter and

QB and all the parallel chords are ordinates to this

diameter.

The proposition of that article may therefore be stated

as follows.

Any diameter of a parabola is parallel to the axis and

the tangent at the point where it Tneets the curve is parallel

to its ordinates.

223. The tangents at the ends of any chord meet on

the diameter which bisects the chord.

Let the equation of QR (Fig., Art. 220) be

y = mx + c (1),

13â€”2

196

COORDINATE GEOMETRY.

and let the tangents at Q and R meet at the point T

Then QR is the chord of contact of tangents drawn

from T^ and hence its equation is

2/2/1 = 2a{x + x^) (Art. 213).

Comparing this with equation (1), we have

2a ,, , 2a

â€” = m, so that Vi = â€” ->

2/1 ^*

and therefore T lies on the straight line

2a

^ m

But this straight line was proved, in Art. 220, to be

the diameter P V which bisects the chord.

224. To find the equation to a parabola, the axes

being any diameter and the tangent to the parabola at the

point where this diameter meets the curve.

Let PVX be the diameter and PY the tangent at P

meeting the axis in T.

Take any point Q on the curve,

and draw QM perpendicular to the

axis meeting the diameter P F in L.

Let PVhQ X and VQ be y.

Draw PN perpendicular to the

axis of the curve, and let

e^ /. YPX=iPTM,

Then

iAS. A]S[^PN^ = ]SfT^ ts,Ti^e=^.AN^ . tan^ 6.

:. ANr=:AS. cot^ e = a cot^ e,

and PN = JIASTaN = 2a cot 6.

Now QM'- = 4:AS.AM=4:a.AM (1).

Also

QM=JSrP + LQ = 2acote+ VQsmO = 2acotO+ysinO,

and AM=A]\/' + PV+ VL=-acot^e + x + ycose.

THE PARABOLA. EXAMPLES. 197

Substituting these values in (1), we have

(2a cot + y sin Oy â€” ia (a cot^ + x + y cos 6),

i. e. if- sin^ 6 â€” ^ax.

The required equation is therefore

y'^^lpx (2),

where

p - T^= Â« (1 + ^ot' Q) = a^ AN= SP (by Art. 202).

The equation to the parabola referred to the above axes

is therefore of the same form as its equation referred to the

rectangular axes of Art. 197.

The equation (2) states that

QV'^^iSP.PV.

225. The quantity 4^j is called the parameter of the

diameter P V. It is equal in length to the chord which is

parallel to P F and passes through the focus.

For if Q'V'R' be the chord, parallel to PZand passing

through the focus and meeting PT in V\ we have

PY' = ST=SP^p,

so that Q' V"" ^ip.PV'^ ip\

and hence Q'R' =-'2Q'V' ^ ip.

226. Just as in Art. 205 it could now be shown that

the tangent at any point {x\ y) of the above curve is

yy â€” 2p (x + x).

Similarly for the equation to the polar of any point.

EXAMPLES. XXVII.

1. Prove that the length of the chord joining the points of

contact of tangents drawn from the point (Xj, y^ is

ijy-^ + 4a2 fjy^^ - 4aa; J ^

a

2. Prove that the area of the triangle formed by the tangents

3

from the point {x^^ y^ and the chord of contact is {y^ - ^ax^^ -^2a.

198 COORDINATE GEOMETRY. [Exs. XXVII.]

3. If a perpendicular be let fall from any point P upon its polar

prove that the distance of the foot of this perpendicular from the

focus is equal to the distance of the point P from the directrix.

4. What is the equation to the chord of the parabola y^ = 8x

which is bisected at the point (2, - 3) ?

5. The general equation to a system of parallel chords in the

parabola y^ = ^x is 4:X-y + k = 0.

Wliat is the equation to the corresponding diameter ?

6. P, Q, and B are three points on a parabola and the chord PQ

cuts the diameter through R in V. Ordinates P3I and QN are drawn

to this diameter. Prove that RM . RN=RV^.

7. Two equal parabolas with axes in opposite directions touch at

a point O. From a point P on one of them are drawn tangents PQ

and PQ' to the other. Prove that QQ' will touch the first parabola in

â– P' where PP' is parallel to the common tangent at O.

Coordinates of any point on the parabola ex-

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