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 E2 passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E2 is

Engineering

Mathematics

Standard Ellipse

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 E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E₂ is

$\frac{3}{4}$

$\frac{1}{2}$

$\frac{\sqrt{3}}{2}$

$\frac{\sqrt{2}}{2}$

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Equation of $E_{2} is \frac{x^{2}}{a^{2}} + \frac{y^{2}}{16} = 1$

Since E₂ passes through (3, 2)

$\Rightarrow \frac{9}{a^{2}} = \frac{3}{4} or a^{2} = 12$

$' e' or E_{2} = \sqrt{1 - \frac{12}{16}} = \frac{1}{2}$

'e' of E₂ = $\sqrt{1 - \frac{12}{16}} = \frac{1}{2}$ . Ans.