Focus-directrix properties and eccentricityEdexcel International A Level Further Maths: Mind map

Exam questions on Focus-directrix properties and eccentricity

1. The ellipse $E$ has equation $\frac{x^2}{25}+\frac{y^2}{16}=1$.
   Find the equations of the directrices of $E$.2 marks
2. The hyperbola $H$ has equation $\frac{x^2}{9}-\frac{y^2}{16}=1$.
   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
3. An ellipse $E$ has its centre at the origin and its foci at $(\pm4,0)$ on the $x$-axis. Its eccentricity is $\frac23$.
   Find the equation of $E$ in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.3 marks