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S. L. (Sidney Luxton) Loney.

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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 —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^ — 4:ax

— 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 — 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 ( — 2 » — ) of the parabola

y^=4iax is

2a . f a\
y . — = 2a[x + ~] ,

d
I.e. ii = mx-\ — .

m

206. To find the condition that the straight line

y = mx + c (1)

rtiay touch the parabola y^ — 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 — 2ay — iirrc^

i.e. ct? - amc — 0,

a
so that G = — .

m

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

equation to a tangent,

a
y = mx + — .
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=—x+^ (1).

y y

In this equation put — , =m, i.e. y =— ,
^ y m

, y'^ a , 2ax' a

and hence x = -r- = ^, , and — — = — .

4a m^ y m

a
The equation (1) then becomes y = mx-] — .

Also it is the tangent at the point {x', y'), i.e. (—^, — 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=z—-{x + x).

Its equation is therefore

y — y = rri' {x — x'),

2fi 11

where m' x — 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 — 2a7n — am^.
In equation (1) of the last article put



2a



= m, %.e. y



- — ^.



am.



/2



Hence



, y 2

X =^ = aiir.
4a



The normal is therefore

y + 2am = m {x



am



%



t.e.



y = mx — 2am — am^



and it is a normal at the point (am^, — 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 —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 — 2am - am^,
i.e. mx-y = 2am+am^.

The parabola is y^ — 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^ — '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 — 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 — 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^ — 12x,

4. at the ends of the latus rectum of the parabola ^2 — 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 60° with
the axis ; find its point of contact.

7. A tangent to the parabola y'^ = 8x makes an angle of 45° 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 30° 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 + —.
' 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^ — 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^ — '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 + ^'^±^-;^^ 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^ — 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 {a« + 6«)

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

y—mx-\ — (Ij.

The equation to the perpendicular to, this line passing
through the focus is

2/ = - (^-«) (2).

The lines (1) and (2) meet where

a \ , . 1 a

nix H — =— —[x— a) = X -^ — 5

7n m m in

i. e. where x — 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 -\ — ( 1 ).

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

a
y, = TUX, + — ,

i. e. m^Xj^ — 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-^ — 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 yi— ^cix^ be zero, i.e. if the point {x-^, y^ lie on
the curve.

The two roots are imaginary if y^ — 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" — 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^ — ^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

^ —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

k— y. hy, - kx.
%.e. y = - — -x^-^ \

If this be a tangent it must be of the form

a

y — mx -{ — ,

, , . k — y^ , hy. — kx. a

so that . — ^ = m and ~- i = — .

a — x^ h — x^ m

Hence, by multiplication,

k — y^ hy^ — kx^

i. e. a (lb - x^^ = {k — y^) [hy^ — 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 - 2« (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 — c),

4:a Aac , .

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

^ m ^ m • '

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" _ 2«

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/ = — meets the curve in a point P,

whose ordinate is — and whose abscissa is therefore — x .
m m"

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

a

y = Tnx -\ — ,

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),

13—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

— = m, so that Vi = — ->
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 — ia (a cot^ + x + y cos 6),

i. e. if- sin^ 6 — ^ax.

The required equation is therefore

y'^^lpx (2),

where

p - T^= « (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 — 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
■P' where PP' is parallel to the common tangent at O.

Coordinates of any point on the parabola ex-


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