Ellipse and hyperbolaEdexcel A-Level Further Maths: Subtopic test

1. 1
   The ellipse $E$ has equation $\frac{x^2}{25}+\frac{y^2}{16}=1$.

   (a)

   Find the eccentricity of $E$.

   [1 mark]

   • A$\frac45$
   • B$\frac{9}{25}$
   • C$\frac35$
   • D$\frac53$

   (b)

   Find the equations of the directrices of $E$.

   [1 mark]

   • A$x=\pm\frac{25}{3}$
   • B$x=\pm3$
   • C$x=\pm\frac95$
   • D$x=\pm\frac{20}{3}$

   (c)

   The point $P$ on $E$ has $x$-coordinate $\frac{10}{3}$. Use the focus-directrix property to find the distance from $P$ to the focus $(3,0)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The hyperbola $H$ has equation $\frac{x^2}{9}-\frac{y^2}{16}=1$, with parametric equations $x=3\sec t$, $y=4\tan t$.

   (a)

   Find the coordinates of the foci of $H$.

   [1 mark]

   • A$(\pm4,0)$
   • B$(\pm5,0)$
   • C$(\pm\sqrt7,0)$
   • D$\left(\pm\frac53,0\right)$

   (b)

   Find the coordinates of the point $P$ on $H$ with $t=\frac{\pi}{3}$.

   [1 mark]

   • A$\left(\frac32,2\sqrt3\right)$
   • B$\left(6,\frac{4}{\sqrt3}\right)$
   • C$\left(3\sqrt3,8\right)$
   • D$\left(6,4\sqrt3\right)$

   (c)

   Find the exact gradient of $H$ at the point $P\left(6,4\sqrt3\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The hyperbola $H$ has equation $\frac{x^2}{4}-\frac{y^2}{9}=1$, with parametric equations $x=2\sec t$, $y=3\tan t$.

   (a)

   The line $y=2x+k$ is a tangent to $H$. Find the possible values of $k$.

   [3 marks]

   (b)

   The point $P$ on $H$ has parameter $t=\frac\pi3$. Find the equation of the tangent to $H$ at $P$ and the coordinates of the point where this tangent meets the $x$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A point $P(x,y)$ moves so that its distance from the point $S(4,0)$ is half its perpendicular distance from the line $l$ with equation $x=16$. The locus of $P$ is the curve $C$.

   (a)

   (i) Show that $C$ has equation $\frac{x^2}{64}+\frac{y^2}{48}=1$.
   (ii) Write down the coordinates of the other focus of $C$ and the equation of the corresponding directrix.

   [6 marks]

   (b)

   The point $Q(4,6)$ lies on $C$. The tangent to $C$ at $Q$ meets the $x$-axis at $A$ and the normal to $C$ at $Q$ meets the $x$-axis at $B$. Show that $A$ lies on $l$ and find the area of triangle $QAB$.

   [6 marks]

   Total for question 4: 12 marks

End of questions