Tangent & Normals to Ellipses, Parabolas & Hyperbolas (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 rectangular hyperbola $H$ has equation $xy=36$

Use calculus to show that the equation of the tangent to $H$ at the point $P\left(6t,\frac{6}{t}\right)$ is

$y{t}^2+x=12t$

The point $Q\left(12t,\frac{3}{t}\right)$ also lies on $H$.

Find the equation of the tangent to $H$ at the point $Q$.

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 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

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

The parabola C has equation

$y^2=16x$

The distinct points $P(p^2,4p)$ and $Q(q^2,4q)$ lie on C, where $p\ne 0$, $q\ne 0$

The tangent to C at P and the tangent to C at Q meet at the point $R(-28,6)$.

Show that the area of triangle PQR is 1331

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.

The normal to the parabola $y^2=4ax$ at the point $P(a{p}^2,2ap)$ passes through the parabola again at the point $Q(a{q}^2,2aq)$.

The line $OP$ is perpendicular to the line $OQ$, where $O$ is the origin.

Prove that $p^2=2$

$P$ and $Q$ are two distinct points on the ellipse described by the equation $x^2+4y^2=4$

The line $l$ passes through the point $P$ and the point $Q$.

The tangent to the ellipse at $P$ and the tangent to the ellipse at $Q$ intersect at the point $(r,s)$.

Show that an equation of the line $l$ is

$4sy+rx=4$

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 $C$.