The ellipse $E_{1}: \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_{2}$ passing through the point $(0, 4)$ circumscribes the rectangle $R$. The eccentricity of the ellipse $E_{2}$ is

The ellipse E₁: x²/9 + y²/4 = 1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E₂ is

$\text{Given: } E_1: \dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$

$\text{inscribed in rectangle } R \ (\text{sides } \parallel \text{ axes})$

$E_2 \text{ passes through } (0,4) \text{ and circumscribes } R$

$E_1: \dfrac{x^2}{9}+\dfrac{y^2}{4}=1 \;\Rightarrow\; a=3,\ b=2$

$\text{vertices } (\pm 3,0),\quad \text{co-vertices } (0,\pm 2)$

$R: -3 \le x \le 3,\quad -2 \le y \le 2$

$\text{corners } (\pm 3,\pm 2)$

$E_2 \ni (\pm 3,\pm 2)\ \text{and}\ (0,4)$

$\text{Plan: } 1.\ E_2 \text{ centred at origin, axes along coordinate axes}$

$2.\ \text{put } (0,4) \text{ and } (3,2) \Rightarrow A,\ B$ $3.\ e=\sqrt{1-\dfrac{(\text{semi-minor})^2}{(\text{semi-major})^2}}$

$E_2: \dfrac{x^2}{A^2}+\dfrac{y^2}{B^2}=1$

$\dfrac{16}{B^2}=1 \;\Rightarrow\; B^2=16 \;\Rightarrow\; B=4$

$\dfrac{9}{A^2}+\dfrac{4}{16}=1$

$\dfrac{9}{A^2}=\dfrac{3}{4}$

$A^2=12$

$E_2: \dfrac{x^2}{12}+\dfrac{y^2}{16}=1$

$B=4 > A=2\sqrt{3} \;\Rightarrow\; \text{major axis along } y$

$e=\sqrt{1-\dfrac{A^2}{B^2}}$

$e=\sqrt{1-\dfrac{12}{16}}=\sqrt{\dfrac{4}{16}}=\dfrac{1}{2}$

$c=Be=4\cdot\dfrac{1}{2}=2\;\Rightarrow\;\text{foci } (0,\pm 2)$

$e=\dfrac{1}{2}\quad\text{option (3)}$