Question

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

• A. $\frac{\sqrt{2}}{2}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{1}{2}$
• D. $\frac{3}{4}$

Answer 1

The correct option is C $\frac{1}{2}$

If an ellipse circumscribes the rectangle, then it must pass through all four vertices.

Let The equation of an ellipse $E_2$ be
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$


It passes through $(0,4).$
$\Rightarrow b^2=16$

Also it passes through $(3,2)$
$\Rightarrow \frac{9}{a^2}+\frac{4}{16}=1\Rightarrow a^2=12$

Eccentricity of $E_2$
$=\sqrt{1-\frac{a^2}{b^2}}=\sqrt{1-\frac{12}{16}}=\frac{1}{2}$