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

• 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 $E$

Answer 1

The correct option is D $P$ lies inside $C$ but outside $E$

For circle $x^2+y^2=9$
point $P$ is $(1,2)$ and $Q$ is $(2,1)$
Since $1^2+2^2-9<0$ and $2^2+1^2-9<0$
$\Rightarrow$ Both $P$ and $Q$ lie inside $C$.

For $\frac{x^2}{9}+\frac{y^2}{4}=1$
Also $\frac{1^2}{9}+\frac{2^2}{4}-1>0$ and $\frac{2^2}{9}+\frac{1^2}{4}-1<0$
$\Rightarrow P$ lies outside $E$ and $Q$ lies inside $E$.
Thus $P$ lies inside $C$ but outside $E.$