question_answer

The ellipse\[{{x}^{2}}+4{{y}^{2}}=4\]is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is     AIEEE  Solved  Paper-2009

A) \[{{x}^{2}}+16{{y}^{2}}=16\]      

B) \[{{x}^{2}}+12{{y}^{2}}=16\]

C)                        \[4{{x}^{2}}+48{{y}^{2}}=48\]

D)        \[4{{x}^{2}}+64{{y}^{2}}=48\]

Correct Answer: B

Solution :

Clearly second ellipse is passing through (4, 0) so semi major axis of the ellipse is ?4?. If length of semi-minor axis is b then \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] It passes through (2, 1) So \[\frac{4}{16}+\frac{1}{{{b}^{2}}}=1\] \[\frac{1}{{{b}^{2}}}=\frac{3}{4}\] \[{{b}^{2}}=\frac{4}{3}\] \[\Rightarrow \] \[{{x}^{2}}+12{{y}^{2}}=16\]