Parabola and rectangular hyperbolaEdexcel A-Level Further Maths: Subtopic test

1. 1
   A parabola $C$ has equation $y^2=20x$.

   (a)

   Find the coordinates of the focus of $C$.

   [1 mark]

   • A$(20,0)$
   • B$(5,0)$
   • C$(10,0)$
   • D$(0,5)$

   (b)

   Find the equation of the directrix of $C$.

   [1 mark]

   • A$x=5$
   • B$y=-5$
   • C$x=-20$
   • D$x=-5$

   (c)

   The point $P$ on $C$ has $x$-coordinate $20$. Use the focus-directrix property to find the distance from $P$ to the focus.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The rectangular hyperbola $H$ has equation $xy=36$.

   (a)

   Which of these points lies on $H$ for every non-zero value of $t$?

   [1 mark]

   • A$(6t,6t)$
   • B$\left(6t,\frac{36}{t}\right)$
   • C$\left(6t,\frac{6}{t}\right)$
   • D$\left(36t,\frac{6}{t}\right)$

   (b)

   Find the gradient of $H$ at the point $(4,9)$.

   [1 mark]

   • A$-\frac94$
   • B$-\frac49$
   • C$\frac94$
   • D$-9$

   (c)

   Find the equation of the normal to $H$ at the point $(4,9)$, giving your answer in the form $ay=bx+c$ where $a$, $b$ and $c$ are integers.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The parabola $C$ has equation $y^2=8x$.

   (a)

   The line $y=mx+2$ is a tangent to $C$. Find the value of $m$.

   [3 marks]

   (b)

   The point $P$ on $C$ has parameter $t=2$. Find the equation of the normal to $C$ at $P$ and the coordinates of the point where this normal meets the $x$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The rectangular hyperbola $H$ has equation $xy=16$, where the point $P\left(4t,\frac{4}{t}\right)$ lies on $H$ with $t>0$.

   (a)

   The tangent to $H$ at $P$ meets the $x$-axis at $A$ and the $y$-axis at $B$, and $O$ is the origin. Find the equation of the tangent at $P$, and hence show that the area of triangle $OAB$ does not depend on $t$.

   [6 marks]

   (b)

   The point $R$ is such that $OARB$ is a rectangle. Find a Cartesian equation for the locus of $R$ as $t$ varies, and describe how this locus is related to $H$.

   [6 marks]

   Total for question 4: 12 marks

End of questions