FP3: Further coordinate systemsEdexcel International A Level Further Maths: Topic test

1. 1
   The ellipse $E$ has parametric equations $x=7\cos t$, $y=4\sin t$, for $0\le t<2\pi$.

   (a)

   Find the Cartesian equation of $E$.

   [1 mark]

   • A$\frac{x^2}{7}+\frac{y^2}{4}=1$
   • B$\frac{x^2}{16}+\frac{y^2}{49}=1$
   • C$\frac{x^2}{49}+\frac{y^2}{16}=1$
   • D$\frac{x^2}{49}-\frac{y^2}{16}=1$

   (b)

   Find the exact $y$-coordinate of the point on $E$ with parameter $t=\frac{\pi}{3}$.

   [1 mark]

   • A$2\sqrt3$
   • B$2$
   • C$4\sqrt3$
   • D$\frac{7\sqrt3}{2}$

   (c)

   Find the two values of $t$, for $0\le t<2\pi$, at which $E$ meets the line $y=2\sqrt2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The hyperbola $H$ has parametric equations $x=2\cosh t$, $y=3\sinh t$, for $t\in\mathbb{R}$.

   (a)

   Find the Cartesian equation of $H$.

   [1 mark]

   • A$\frac{x^2}{4}+\frac{y^2}{9}=1$
   • B$\frac{x^2}{2}-\frac{y^2}{3}=1$
   • C$\frac{x^2}{9}-\frac{y^2}{4}=1$
   • D$\frac{x^2}{4}-\frac{y^2}{9}=1$

   (b)

   Find the equations of the asymptotes of $H$.

   [1 mark]

   • A$y=\pm\frac23x$
   • B$y=\pm\frac32x$
   • C$y=\pm\frac94x$
   • D$y=\pm3x$

   (c)

   Find the exact coordinates of the point on $H$ for which $t=\ln2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The ellipse $E$ has equation $\frac{x^2}{16}+\frac{y^2}{7}=1$ and has foci $S$ and $S'$, where $S$ has positive $x$-coordinate.

   (a)

   Find the eccentricity of $E$ and the coordinates of $S$ and $S'$.

   [3 marks]

   (b)

   Write down the equations of the directrices of $E$. The point $P(0,\sqrt7)$ lies on $E$. Verify that $PS=e\times$ (the perpendicular distance from $P$ to the directrix $x=\frac{16}{3}$).

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The hyperbola $H$ has equation $\frac{x^2}{20}-\frac{y^2}{5}=1$. The point $P$ lies on $H$ and has $x$-coordinate $6$ and positive $y$-coordinate.

   (a)

   Find the eccentricity of $H$, the coordinates of its foci and the equations of its directrices. Show that $P$ satisfies the focus-directrix property for the focus with positive $x$-coordinate.

   [6 marks]

   (b)

   Find an equation of the normal to $H$ at $P$. This normal meets the $x$-axis at $Q$. Given that $S$ is the point $(5,0)$, find the area of triangle $SPQ$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The ellipse $E$ has equation $\frac{x^2}{8}+\frac{y^2}{2}=1$.

   (a)

   The line $y=mx+c$ is a tangent to $E$. Which of the following conditions must hold?

   [1 mark]

   • A$c^2=2m^2+8$
   • B$c^2=8m^2+2$
   • C$c^2=8m^2-2$
   • D$c=8m+2$

   (b)

   Find the gradient of the tangent to $E$ at the point $(2,1)$.

   [1 mark]

   • A$2$
   • B$\frac12$
   • C$-2$
   • D$-\frac12$

   (c)

   Find the values of $c$ for which the line $y=-x+c$ is a tangent to $E$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   An ellipse $E$ has its centre at the origin and its foci on the $x$-axis at $(\pm2,0)$. The directrices of $E$ are $x=\pm8$.

   (a)

   Find the semi-major axis $a$ of $E$.

   [1 mark]

   • A$4$
   • B$2$
   • C$8$
   • D$16$

   (b)

   Find the eccentricity of $E$.

   [1 mark]

   • A$2$
   • B$\frac14$
   • C$\frac12$
   • D$4$

   (c)

   Find the Cartesian equation of $E$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The point $P(x,y)$ moves in the plane so that its distance from the point $A(-3,0)$ is twice its distance from the point $B(3,0)$.

   (a)

   Show that the locus of $P$ is a circle, and state its centre and radius.

   [3 marks]

   (b)

   The two tangents to the locus from the origin are drawn. Find their equations.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The ellipse $E$ has parametric equations $x=6\cos t$, $y=3\sin t$, for $0\le t<2\pi$. The point $P$ on $E$ has parameter $t$, where $0<t<\frac{\pi}{2}$.

   (a)

   Show that an equation of the tangent to $E$ at $P$ is $x\cos t+2y\sin t=6$. Hence find an equation of the tangent to $E$ at the point where $t=\frac{\pi}{3}$.

   [6 marks]

   (b)

   The tangent at $P$ meets the $x$-axis at $A$ and the $y$-axis at $B$, and $M$ is the midpoint of $AB$. Find the coordinates of $M$ in terms of $t$, and show that, as $t$ varies, $M$ lies on the curve with equation $\frac{9}{x^2}+\frac{9}{4y^2}=1$.

   [6 marks]

   Total for question 8: 12 marks

End of questions