First slide

Ellipse

Question

$Let E be the ellipse \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 and C be the circle x^{2} + y^{2} = 9 . Let P and Q be the points (1, 2) and (2, 1) respectively. Then,$

Moderate

A

Q lies inside C but outside E
B

Q lies outside both C and E
C

P lies inside both C and E
D

P lies inside C but outside E

Solution

$Since 1^{2} + 2^{2} = 5 < 9 and 2^{2} + 1^{2} = 5 < 9, both P and Q lie inside C . Also,$
$\frac{1^{2}}{9} + \frac{2^{2}}{4} = \frac{1}{9} + 1 > 1 and \frac{2^{2}}{9} + \frac{1}{4} = \frac{25}{36} < 1$
Hence, P lies outside E and Q lies inside E.
Thus, P lies inside C but outside E.

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