Properties of Ellipses, Parabolas & Hyperbolas (Edexcel A Level Further Maths: Further Pure 1): Exam Questions

The parabola $P$ has equation $y^2=4ax$, where $a$ is a positive constant.

The point $A\left(a{t}^2,2at\right)$, where $t\ne 0$, lies on $P$.

Use calculus to show that an equation of the tangent to $P$ at $A$ is

$yt=x+a{t}^2$

The point $B\left(2k^2,4k\right)$ and the point $C\left(2k^2,-4k\right)$, where $k$ is a constant, lie on $P$.

The tangent to $P$ at $B$ and the tangent to $P$ at $C$ intersect at the point $D$.

Given that the area of the triangle $BCD$ is 432, determine the coordinates of $B$ and the coordinates of $C$.

The ellipse $E$ has equation

$\frac{x^2}{16}+\frac{y^2}{9}=1$

Determine the exact value of the eccentricity of $E$

The points $P(4\text{cos}\theta ,3\text{sin}\theta )$ and $Q(4\text{cos}\theta ,-3\text{sin}\theta )$ lie on $E$ where $0<\theta <\frac{\pi }{2}$

The line $l_1$ is the normal to $E$ at the point $P$

Use calculus to show that $l_1$ has equation

$4x\text{sin}\theta -3y\text{cos}\theta =7\text{sin}\theta \text{cos}\theta$

The line $l_2$ passes through the origin and the point $Q$

The lines $l_1$ and $l_2$ intersect at the point $R$

Determine, in simplest form, the coordinates of $R$

Hence show that, as $\theta$ varies, $R$ lies on an ellipse which has the same eccentricity as ellipse $E$

An ellipse has equation $\frac{x^2}{16}+\frac{y^2}{4}=1$ and eccentricity $e_1$

A hyperbola has equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and eccentricity $e_2$

Given that $e_1\times e_2=1$

show that $a^2=3b^2$

Given also that the coordinates of the foci of the ellipse are the same as the coordinates of the foci of the hyperbola,

determine the equation of the hyperbola.

The parabola C has equation $y^2=32x$

and the hyperbola H has equation $\frac{x^2}{36}-\frac{y^2}{9}=1$

Write down the equations of the asymptotes of H.

The line $l_1$ is normal to C and parallel to the asymptote of H with positive gradient.

The line $l_2$ is normal to C and parallel to the asymptote of H with negative gradient.

Determine

(i) an equation for $l_1$

(ii) an equation for $l_2$

The lines $l_1$ and $l_2$ meet H at the points P and Q respectively.

Find the area of the triangle OPQ, where O is the origin.

The ellipse E has equation

$\frac{x^2}{36}+\frac{y^2}{20}=1$

Find the coordinates of the foci of E.

Find the equations of the directrices of E.

The ellipse E has equation

$\frac{x^2}{36}+\frac{y^2}{16}=1$

The points S and S' are the foci of E.

Find the coordinates of S and S'

Show that for any point P on E, the triangle PSS' has constant perimeter and determine its value.

The hyperbola $H$ has equation

$\frac{x^2}{16}-\frac{y^2}{9}=1$

The ellipse $E$ has equation

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

where $a$ and $b$ are positive constants.

Given that

• the eccentricity of $E$ is the reciprocal of the eccentricity of $H$

• the coordinates of the foci of $E$ are the same as the coordinates of the foci of $H$

determine

(i) the value of $a$

(ii) the value of $b$

The hyperbola H has equation

$\frac{x^2}{16}-\frac{y^2}{9}=1$

The line $l_1$ is the tangent to H at the point $P(4\text{cosh}\theta ,3\text{sinh}\theta )$.

The line $l_1$ meets the $x$-axis at the point A.

The line $l_2$ is the tangent to H at the point $(4,0)$.

The lines $l_1$ and $l_2$ meet at the point B and the midpoint of AB is the point M.

Show that, as $\theta$ varies, a Cartesian equation for the locus of M is

$y^2=\frac{9(4-x)}{4x}p<x<q$

where $p$ and $q$ are values to be determined.

Let S be the focus of H that lies on the positive $x$-axis.

Show that the distance from M to S is greater than 1

The parabola $P$ has equation $y^2=4ax$, where $a$ is a positive constant.

The point $A(a{t}^2,2at)$, where $t\ne 0$, lies on $P$.

Use calculus to show that an equation of the normal to $P$ at $A$ is

$y+tx=2at+a{t}^3$

The point $B(3k^2,6k)$ and the point $C(3k^2,-6k)$, where $k$ is a constant, lie on $P$.

The normal to $P$ at $B$ and the normal to $P$ at $C$ intersect at the point $D$.

Given that the area of triangle $BCD$ is 108, determine the coordinates of $B$ and the coordinates of $C$.