The ellipse and hyperbolaEdexcel International A Level Further Maths: Subtopic test

1. 1
   The ellipse $E$ has parametric equations $x=5\cos t$, $y=3\sin t$, for $0\le t<2\pi$.

   (a)

   Which is a Cartesian equation of $E$?

   [1 mark]

   • A$\frac{x^2}{9}+\frac{y^2}{25}=1$
   • B$\frac{x^2}{25}+\frac{y^2}{9}=1$
   • C$5x^2+3y^2=1$
   • D$\frac{x^2}{5}+\frac{y^2}{3}=1$

   (b)

   Find the coordinates of the point on $E$ where $t=\frac{\pi}{3}$.

   [1 mark]

   • A$\left(\frac52,\frac{3\sqrt3}{2}\right)$
   • B$\left(\frac{5\sqrt3}{2},\frac32\right)$
   • C$\left(\frac52,\frac32\right)$
   • D$\left(\frac32,\frac{5\sqrt3}{2}\right)$

   (c)

   Find the value of $t$, for $0\le t<2\pi$, at the point $\left(4,-\frac95\right)$ on $E$. Give your answer to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The hyperbola $H$ has Cartesian equation $\frac{x^2}{9}-\frac{y^2}{16}=1$.

   (a)

   Which pair of parametric equations satisfies the equation of $H$?

   [1 mark]

   • A$x=3\cos t$, $y=4\sin t$
   • B$x=4\sec t$, $y=3\tan t$
   • C$x=3\sec t$, $y=4\tan t$
   • D$x=3\sinh t$, $y=4\cosh t$

   (b)

   A point of $H$ has coordinates $(3\cosh t,4\sinh t)$. Find its coordinates when $t=\ln2$.

   [1 mark]

   • A$(6,8)$
   • B$\left(\frac{15}{4},5\right)$
   • C$\left(\frac94,5\right)$
   • D$\left(\frac{15}{4},3\right)$

   (c)

   Show that $x=3\cosh t$, $y=4\sinh t$ satisfies the Cartesian equation of $H$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The ellipse $E$ has Cartesian equation $\frac{x^2}{16}+\frac{y^2}{4}=1$.

   (a)

   Write down parametric equations for $E$. Hence find the exact value of the parameter $t$, for $0<t<\frac{\pi}{2}$, at the point $\left(2\sqrt2,\sqrt2\right)$ on $E$.

   [3 marks]

   (b)

   The point $P$ lies on $E$. Using a parametrisation of $E$, find the greatest and least values of the distance $OP$, where $O$ is the origin.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The hyperbola $H$ has parametric equations $x=4\sec t$, $y=3\tan t$, and has asymptotes $y=\pm\frac34x$. The point $P$ on $H$ has parameter $t=\frac{\pi}{3}$.

   (a)

   Show that a Cartesian equation of $H$ is $\frac{x^2}{16}-\frac{y^2}{9}=1$. Find the coordinates of $P$, and find the exact value of $u$ for which $P$ is the point $(4\cosh u,3\sinh u)$.

   [6 marks]

   (b)

   The line $L$ has equation $3x-4y=12$. Show that $L$ is parallel to an asymptote of $H$, and find the coordinates of the point $Q$ where $L$ meets $H$.

   [6 marks]

   Total for question 4: 12 marks

End of questions