FP1: Coordinate systemsEdexcel International A Level Further Maths: Topic test

1. 1
   The parabola $C$ has equation $y^2=24x$. Its focus is $S$ and its directrix is the line $l$.

   (a)

   Find the coordinates of $S$.

   [1 mark]

   • A$(12,0)$
   • B$(6,0)$
   • C$(24,0)$
   • D$(0,6)$

   (b)

   Find the equation of $l$.

   [1 mark]

   • A$x=6$
   • B$x=-12$
   • C$x=-6$
   • D$y=-6$

   (c)

   The point $P(6,12)$ lies on $C$. Verify that $P$ is equidistant from $S$ and $l$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The rectangular hyperbola $H$ has parametric equations $x=7t$, $y=\dfrac7t$, where $t\neq0$.

   (a)

   Find the Cartesian equation of $H$.

   [1 mark]

   • A$xy=49$
   • B$xy=7$
   • C$xy=14$
   • D$x+y=14$

   (b)

   Find the coordinates of the point on $H$ where $t=-2$.

   [1 mark]

   • A$\left(14,\dfrac72\right)$
   • B$\left(-14,\dfrac72\right)$
   • C$\left(-\dfrac72,-14\right)$
   • D$\left(-14,-\dfrac72\right)$

   (c)

   The line $y=x$ meets $H$ at two points. Find their coordinates.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The parabola $C$ has equation $y^2=32x$. The point $P(2,8)$ lies on $C$ and the focus of $C$ is $S$.

   (a)

   Find the equation of the tangent to $C$ at $P$, giving your answer in the form $y=mx+c$.

   [3 marks]

   (b)

   The normal to $C$ at $P$ meets the $x$-axis at $Q$. Find the coordinates of $Q$ and the area of triangle $PSQ$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The rectangular hyperbola $H$ has equation $xy=100$. The point $P(20,5)$ lies on $H$.

   (a)

   The tangent to $H$ at $P$ meets the $x$-axis at $A$ and the $y$-axis at $B$. Find the equation of the tangent, the coordinates of $A$ and $B$, and the area of triangle $OAB$, where $O$ is the origin.

   [6 marks]

   (b)

   The normal to $H$ at $P$ meets $H$ again at the point $Q$. Find the coordinates of $Q$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The parabola $C$ has equation $y^2=28x$. A general point on $C$ has coordinates $(7t^2,14t)$, where $t$ is a parameter, and the focus of $C$ is $S$.

   (a)

   Find the coordinates of the point $P$ on $C$ where $t=2$.

   [1 mark]

   • A$(14,28)$
   • B$(28,14)$
   • C$(14,14)$
   • D$(28,28)$

   (b)

   The point $P$ is the point where $t=2$. Find the distance $PS$.

   [1 mark]

   • A$35$
   • B$21$
   • C$28$
   • D$28\sqrt2$

   (c)

   Show that the point $Q(63,42)$ lies on $C$ and find the value of $t$ at $Q$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The rectangular hyperbola $H$ has parametric equations $x=8t$, $y=\dfrac8t$, where $t\neq0$.

   (a)

   Find the coordinates of the point on $H$ where $t=\dfrac14$.

   [1 mark]

   • A$(32,2)$
   • B$(2,32)$
   • C$(2,2)$
   • D$(-2,-32)$

   (b)

   The line $y=2x$ meets $H$ at two points. Find the positive $x$-coordinate of the point of intersection.

   [1 mark]

   • A$4$
   • B$8\sqrt2$
   • C$4\sqrt2$
   • D$32$

   (c)

   Find the gradient of $H$ at the point $P(2,32)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The parabola $C$ has equation $y^2=40x$. The point $P(10,20)$ lies on $C$.

   (a)

   Find the equation of the normal to $C$ at $P$.

   [3 marks]

   (b)

   The normal to $C$ at $P$ meets $C$ again at the point $Q$. Find the coordinates of $Q$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The parabola $C$ has equation $y^2=36x$ and the rectangular hyperbola $H$ has equation $xy=48$. The curves meet at the point $P(4,12)$. The focus of $C$ is $S$ and its directrix is the line $l$.

   (a)

   (i) Write down the coordinates of $S$ and the equation of $l$.
   (ii) Find the equation of the tangent to $C$ at $P$, and show that it meets the $x$-axis at $T(-4,0)$.
   (iii) Show that $SP=ST$.

   [6 marks]

   (b)

   The tangent to $H$ at $P$ meets the $x$-axis at $A$ and the $y$-axis at $B$. Find the equation of this tangent and show that $P$ is the mid-point of $AB$.

   [6 marks]

   Total for question 8: 12 marks

End of questions