Conic sectionsAQA A-Level Further Maths: Subtopic test

1. 1
   An ellipse $E$ has equation $\dfrac{x^2}{25}+\dfrac{y^2}{9}=1$.

   (a)

   Which pair of points are where $E$ meets the $x$-axis?

   [1 mark]

   • A$(25,0)$ and $(-25,0)$
   • B$(3,0)$ and $(-3,0)$
   • C$(5,0)$ and $(-5,0)$
   • D$(\sqrt5,0)$ and $(-\sqrt5,0)$

   (b)

   Which of these points lies on $E$?

   [1 mark]

   • A$\left(4,\frac95\right)$
   • B$(4,3)$
   • C$(5,3)$
   • D$\left(4,\frac35\right)$

   (c)

   The line $y=\frac95$ meets $E$ at two points. Find the distance between them.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A parabola $P$ has equation $y^2=12x$.

   (a)

   Which of these points lies on $P$?

   [1 mark]

   • A$(6,3)$
   • B$(12,3)$
   • C$(3,12)$
   • D$(3,6)$

   (b)

   Which line is a line of symmetry of $P$?

   [1 mark]

   • A$x=0$
   • B$y=0$
   • C$y=x$
   • D$x=3$

   (c)

   Find the coordinates of the points where $P$ meets the line $y=2x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A hyperbola $H$ has equation $\dfrac{x^2}{16}-\dfrac{y^2}{9}=1$.

   (a)

   Write down the coordinates of the points where $H$ meets the $x$-axis, and show that $H$ does not meet the $y$-axis.

   [3 marks]

   (b)

   (i) Find the equations of the asymptotes of $H$.
   (ii) Find the coordinates of the points on $H$ where $x=5$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The rectangular hyperbola $H$ has equation $xy=9$.

   (a)

   (i) Write down the equations of the asymptotes of $H$.
   (ii) Show that the line $x+y=10$ meets $H$ at $(1,9)$ and $(9,1)$.
   (iii) Find the distance between these two points.

   [6 marks]

   (b)

   (i) Show that the line $x+y=k$ meets $H$ at two distinct points only when $k<-6$ or $k>6$.
   (ii) Hence find the values of $k$ for which the line meets $H$ at exactly one point, and the coordinates of that point in each case.

   [6 marks]

   Total for question 4: 12 marks

End of questions