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 $$P = \left( {1,\,2} \right)$$ and $$Q = \left( {2,\,1} \right).$$ Which one of the following is correct ?
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 :
$$P$$ lies inside $$C$$ but outside $$E$$
Solution :
Given equation of ellipse $$E$$ is
$$\eqalign{
& \frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1 \cr
& \Rightarrow \frac{{4{x^2} + 9{y^2}}}{{36}} = 1 \cr
& \Rightarrow 4{x^2} + 9{y^2} = 36 \cr
& \Rightarrow 4{x^2} + 9{y^2} - 36 = 0......\left( 1 \right) \cr} $$
And $$C\,:$$ equation of circle is $${x^2} + {y^2} = 9$$
Which can be rewritten as $${x^2} + {y^2} - 9 = 0......\left( 2 \right)$$
For a point $$P = \left( {1,\,2} \right)$$ we have
$$\eqalign{
& 4{\left( 1 \right)^2} + 9{\left( 2 \right)^2} - 36 = 40 - 36 > 0\,\,\,\,\left[ {{\text{from equation }}\left( 1 \right)} \right] \cr
& \,{\text{and }}\,{1^2} + {2^2} - 9 = 5 - 9 < 0\,\,\,\,\,\left[ {{\text{from equation }}\left( 2 \right)} \right] \cr} $$
$$\therefore $$ Point $$P$$ lies outside of $$E$$ and inside of $$C.$$