Focus-directrix properties and eccentricityEdexcel International A Level Further Maths: Subtopic test

1. 1
   The ellipse $E$ has equation $\frac{x^2}{25}+\frac{y^2}{16}=1$.

   (a)

   Find the eccentricity of $E$.

   [1 mark]

   • A$\frac45$
   • B$\frac{9}{25}$
   • C$\frac34$
   • D$\frac35$

   (b)

   Find the coordinates of the foci of $E$.

   [1 mark]

   • A$(\pm4,0)$
   • B$(\pm3,0)$
   • C$(0,\pm3)$
   • D$(\pm5,0)$

   (c)

   Find the equations of the directrices of $E$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The hyperbola $H$ has equation $\frac{x^2}{9}-\frac{y^2}{16}=1$.

   (a)

   Find the eccentricity of $H$.

   [1 mark]

   • A$\frac53$
   • B$\frac43$
   • C$\frac54$
   • D$\frac{25}{9}$

   (b)

   Find the coordinates of the foci of $H$.

   [1 mark]

   • A$(\pm4,0)$
   • B$(\pm7,0)$
   • C$(\pm5,0)$
   • D$(0,\pm5)$

   (c)

   The point $P\left(5,\frac{16}{3}\right)$ lies on $H$. Verify that $PS=e\times PM$, where $S$ is the focus $(5,0)$ and $M$ is the foot of the perpendicular from $P$ to the corresponding directrix.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An ellipse $E$ has its centre at the origin and its foci at $(\pm4,0)$ on the $x$-axis. Its eccentricity is $\frac23$.

   (a)

   Find the equation of $E$ in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

   [3 marks]

   (b)

   Find the equations of the directrices of $E$. The point $P\left(3,\sqrt{15}\right)$ lies on $E$. Show that $PS=e\times PM$, where $S$ is the focus $(4,0)$ and $M$ is the foot of the perpendicular from $P$ to the directrix $x=\frac ae$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A point $P$ moves so that its distance from the point $S(6,0)$ is $\frac32$ times its distance from the line $x=\frac83$. The locus of $P$ is the curve $C$.

   (a)

   Show that $C$ has equation $\frac{x^2}{16}-\frac{y^2}{20}=1$.

   [6 marks]

   (b)

   Use $b^2=a^2\left(e^2-1\right)$ to find the eccentricity of $C$, and write down the coordinates of its foci and the equations of its directrices. The point $Q\left(8,2\sqrt{15}\right)$ lies on $C$. Show that $Q$ satisfies the focus-directrix property for the focus $(-6,0)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions