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 two points $(1,2)$ and $(2,1)$ respectively. Then:

• A. $Q$ lies inside $C$ but outside $E$
• B. $Q$ lies outside of both $C$ and $E$
• C. $P$ lies inside of 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 $E$

Equation of ellipse is $\frac{x^2}{9}+\frac{y^2}{4}-1=0$

Equation of circle is $x^2+y^2-9=0$

Substituting the coordinates of $P(1,2)$ in the equation of ellipse, we get

$\frac{1}{9}+1-1>0$

Similarly, substituting in the equation of circle, we get

$1+4-9<0$

$\therefore$ $P$ is inside the circle and outside the ellipse.

Similarly verifying for $Q(2,1)$

$\frac{4}{9}+\frac{1}{4}-1<0$

$1+4-9<0$

$\therefore Q$ lies inside both circle and ellipse.