The parabolaEdexcel International A Level Further Maths: Subtopic test

1. 1
   The parabola $C$ has equation $y^2=12x$. Its focus is $S$.

   (a)

   Find the coordinates of $S$.

   [1 mark]

   • A$(12,0)$
   • B$(6,0)$
   • C$(3,0)$
   • D$(0,3)$

   (b)

   Find the equation of the directrix of $C$.

   [1 mark]

   • A$y=-3$
   • B$x=3$
   • C$x=-12$
   • D$x=-3$

   (c)

   The point $P(12,12)$ lies on $C$. Use the focus-directrix property to find the distance $SP$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The parabola $C$ has equation $y^2=8x$. A general point on $C$ has coordinates $(2t^2,4t)$, where $t$ is a parameter.

   (a)

   Find the coordinates of the point on $C$ with parameter $t=-3$.

   [1 mark]

   • A$(18,12)$
   • B$(18,-12)$
   • C$(6,-12)$
   • D$(-18,-12)$

   (b)

   Which of these points lies on $C$?

   [1 mark]

   • A$(8,8)$
   • B$(8,4)$
   • C$(2,8)$
   • D$(4,16)$

   (c)

   The point $Q(50,-20)$ lies on $C$. Find the value of $t$ at $Q$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The parabola $C$ has equation $y^2=16x$, focus $S$ and directrix $l$. The point $P(4t^2,8t)$ lies on $C$.

   (a)

   Show that the distance $SP$ is equal to the perpendicular distance from $P$ to the directrix.

   [3 marks]

   (b)

   Given that $SP=20$, find the coordinates of the possible positions of $P$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The parabola $C$ has equation $y^2=20x$, focus $S$ and directrix $l$, and $O$ is the origin.

   (a)

   Given that $P$ lies on $C$ in the first quadrant and $SP=30$,
   (i) find the coordinates of $P$,
   (ii) find the exact area of triangle $OSP$.

   [6 marks]

   (b)

   (i) A point $Q(x,y)$ is equidistant from $S$ and from $l$. Show that $Q$ lies on $C$.
   (ii) Determine whether the point $T(10,15)$ is nearer to $S$ or to $l$, and state whether $T$ lies on $C$.

   [6 marks]

   Total for question 4: 12 marks

End of questions