Question

Sol. $\frac{\left( \frac{x - 2y + 1}{\sqrt{5}} \right)^{2}}{(5/4)} + \frac{\left( \frac{2x + y + 2}{\sqrt{5}} \right)^{2}}{(5/9)} = 1$

$$\begin{matrix} & & \ \ \ \ \ \ \ \ & b^{2} & \ = a^{2}(1 - e)^{2} \\ \Rightarrow & & \ \ \ \ \ \ \ \ & \frac{5}{9} & \ = \frac{5}{4}\left( 1 - e^{2} \right) \\ \Rightarrow & & \ \ \ \ \ \ \ \ & e & \ = \frac{\sqrt{5}}{3} \end{matrix}$$

Intersection of major axis which is $2x + y + 2 = 0$ and minor axis which is $x - 2y + 1 = 0$ is $( - 1,0)$

⇒ centre $\equiv ( - 1,0)$

Length of major axis $= 2a = \sqrt{5}$