The ellipse is inscribed in a rectangle whose sides are parallel to the coordinate axes. Another ellipse passing through the point (0, 4) circumscribes the rectangle . The eccentricity of the ellipse is [2012]

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 [2012]

Smart Achievers · Accepted Answer

(3)

$As rectangle A B C D circumscribes the ellipse \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1,$

$R$ $A = (3, 2)$

$Let the ellipse circumscribing the rectangle A B C D is$ $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 . . . (i)$

$Given that ellipse (i) passes through (a, 4) \Rightarrow b^{2} = 16$

$Also ellipse (i) passes through A (3, 2)$

$∴$ $\frac{9}{a^{2}} + \frac{4}{16} = 1 \Rightarrow \frac{9}{a^{2}} = 1 - \frac{1}{4} = \frac{3}{4}$

$\Rightarrow a^{2} = 12$

$∴$ $e = \sqrt{1 - \frac{b^{2}}{a^{2}}} = \sqrt{1 - \frac{12}{16}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$

Solution

Q.
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 [2012]

1 $\frac{\sqrt{2}}{2}$

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

3 $\frac{1}{2}$

4 $\frac{3}{4}$

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Solution

Q. The ellipse E1:x29+y24=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 [2012]

1 22

2 32

3 12