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Q.

An ellipse is drawn by taking a diameter of the circle ${(x - 1)}^{2} + y^{2} = 1$ as its semi minor axis and a diameter of the circle $x^{2} + {(y - 2)}^{2} = 4$ as its semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is

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a

$x^{2} + 4 y^{2} = 16$

b

$4 x^{2} + y^{2} = 4$

c

$x^{2} + y^{2} = 8$

d

$4 x^{2} + y^{2} = 8$

answer is D.

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Detailed Solution

Given circles are

${(x - 1)}^{2} + y^{2} = 1 \Rightarrow l e n g t h o f s e m i m i n o r a x i s = d i a m e t e r o f c i r c l e = 2 r = 2 i . e b = 2 a n d x^{2} + {(y - 2)}^{2} = 4 \Rightarrow l e n g t h o f m a j o r a x i s = l e n g t h o f d i a m e t e r = 2 r = 4 i . e a = 4$

$∴ eq u a t i o n of ellipse i s \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \Rightarrow \frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$

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