Loci Problems (Edexcel A Level Further Maths: Further Pure 1): Exam Questions

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 $E$

The points $P(9p^2,18p)$ and $Q(9q^2,18q)$, $p\ne q$, lie on the parabola C with equation

$y^2=36x$

The line $l$ passes through the points P and Q

Show that an equation for the line $l$ is

$(p+q)y=2(x+9pq)$

The normal to C at P and the normal to C at Q meet at the point A.

Show that the coordinates of A are

$\left(9\right(p^2+q^2+pq+2),-9pq(p+q\left)\right)$

Given that the points P and Q vary such that $l$ always passes through the point $(12,0)$

find, in the form $y^2=\text{f}\left(x\right)$, an equation for the locus of A, giving $\text{f}\left(x\right)$ in simplest form.

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.