Conic Sections — Practice

Question 1

Find the focus, directrix and latus rectum of $x^{2}=-12y$.

Accepted Answer

$x^{2}=-4ay$ with $4a=12$, so $a=3$.

Focus $(0,-3)$, directrix $y=3$, latus rectum $12$.

Question 2

The ends of a focal chord of $y^{2}=16x$ are at parameters $t$ and $-	frac1t$. If one end is $(4,8)$, find the other.

Accepted Answer

$4a=16$ so $a=4$, and the point is $\left(4t^{2},8tight)$.

Matching $(4,8)$ gives $t=1$.

The other parameter is $-1$, giving the point $(4,-8)$.

This is the latus rectum, of length $16=4a$.

Question 3

Find the length of the focal chord of $y^{2}=4ax$ that makes an angle of $60^{\circ}$ with the axis.

Accepted Answer

For ends $t_1,t_2$ the squared length is $a^{2}\left(t_1-t_2ight)^{2}\left[\left(t_1+t_2ight)^{2}+4ight]$.

With $t_1t_2=-1$, $\left(t_1-t_2ight)^{2}=\left(t_1+t_2ight)^{2}+4$, so the length is $a\left[\left(t_1+t_2ight)^{2}+4ight]$.

The slope is $\dfrac{2}{t_1+t_2}=	an	heta$, so $t_1+t_2=2\cot	heta$ and the length becomes $4a\csc^{2}	heta=\dfrac{4a}{\sin^{2}	heta}$.

At $	heta=60^{\circ}$ this gives $\dfrac{16a}{3}$, and at $	heta=90^{\circ}$ it gives $4a$, the latus rectum.

Question 4

Show that the normals at three points of $y^{2}=4ax$ are concurrent only if the sum of the parameters is zero, and find the third parameter if two are $1$ and $2$.

Accepted Answer

The cubic $at^{3}+(2a-h)t+k=0$ has no $t^{2}$ term, so $t_1+t_2+t_3=0$.

With $t_1=1$ and $t_2=2$, the third is $t_3=-3$.

Question 5

Find the eccentricity of the ellipse whose foci are at $(\pm2,0)$ and which passes through $(2,3)$.

Accepted Answer

$SP+S'P=2a$ with $S=(2,0)$ and $S'=(-2,0)$.

$SP=3$ and $S'P=\sqrt{16+9}=5$, so $2a=8$ and $a=4$.

$ae=2$ gives $e=	frac12$.

Question 6

The eccentric angles of two points on $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ differ by $\dfrac{\pi}{2}$. Show that the chord joining them touches a fixed ellipse.

Accepted Answer

The chord joining $	heta$ and $\phi$ is $\dfrac xa\cos\dfrac{	heta+\phi}{2}+\dfrac yb\sin\dfrac{	heta+\phi}{2}=\cos\dfrac{	heta-\phi}{2}$.

With $	heta-\phi=	frac{\pi}{2}$ the right side is $\dfrac{1}{\sqrt2}$.

Dividing through, the chord always touches $\dfrac{x^{2}}{a^{2}/2}+\dfrac{y^{2}}{b^{2}/2}=1$.

Question 7

Find the equation of the tangent to $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ that is parallel to $2x+3y=7$.

Accepted Answer

Slope $m=-	frac23$, and $y=mx\pm\sqrt{9m^{2}+4}$.

$9\cdot	frac49+4=8$, so the tangents are $y=-	frac23x\pm2\sqrt2$.

Equivalently $2x+3y=\pm6\sqrt2$.

Question 8

Identify the conic $9x^{2}+4y^{2}-18x+16y-11=0$ and find its centre and eccentricity.

Accepted Answer

$9\left(x-1ight)^{2}+4\left(y+2ight)^{2}=11+9+16=36$.

$\dfrac{(x-1)^{2}}{4}+\dfrac{(y+2)^{2}}{9}=1$: an ellipse centred at $(1,-2)$ with the major axis vertical.

Here $a^{2}=9$, $b^{2}=4$, so $e=\sqrt{1-	frac49}=\dfrac{\sqrt5}{3}$.

Question 9

Find the eccentricity of the hyperbola whose asymptotes are $y=\pm\dfrac34x$.

Accepted Answer

$\dfrac ba=\dfrac34$.

$e^{2}=1+\dfrac{b^{2}}{a^{2}}=1+\dfrac{9}{16}=\dfrac{25}{16}$.

$e=\dfrac54$.

Question 10

A tangent to $xy=16$ meets the axes at $A$ and $B$. Find the area of triangle $OAB$.

Accepted Answer

With $c^{2}=16$, $c=4$, and the tangent at $\left(4t,	frac4tight)$ is $x+t^{2}y=8t$.

Intercepts: $A=(8t,0)$ and $B=\left(0,	frac8tight)$.

Area $=	frac12\cdot8t\cdot	frac8t=32=2c^{2}$, the same for every tangent.

Question 11

Find the chord of the hyperbola $\dfrac{x^{2}}{9}-\dfrac{y^{2}}{4}=1$ whose midpoint is $(5,2)$.

Accepted Answer

$T=\dfrac{5x}{9}-\dfrac{2y}{4}-1$ and $S_1=\dfrac{25}{9}-\dfrac44-1$.

$\dfrac{5x}{9}-\dfrac y2=\dfrac{25}{9}-1$, that is $\dfrac{5x}{9}-\dfrac y2=\dfrac{16}{9}$.

Multiplying by $18$: $10x-9y=32$.

Check: $(5,2)$ gives $50-18=32$.

Question 12

From the point $(1,4)$, find the chord of contact of the tangents to $x^{2}+4y^{2}=16$, after checking that the point is outside.

Accepted Answer

$S_1=1+64-16=49>0$, so the point is outside and the tangents are real.

$T=xx_1+4yy_1-16=x+16y-16$.

Chord of contact: $x+16y=16$.

Practice — Conic Sections

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