Question

If a and $c$ are positive real numbers and the ellipse $\frac{x^2}{4 c^2}+\frac{y^2}{c^2}=1$ has four distinct points in common with the circle $x^2+y^2=9 a^2$, then

• A. $9 a c-9 a^2-2 c^2>0$
• B. $6 ac +9 a ^2-2 c ^2<0$
• C. $6 ac +9 a ^2-2 c ^2>0$
• D. $9 ac -9 a ^2-2 c ^2<0$

Answer 1

Answer: (A)

$ \text { Here, } c<3 a<2 c $
$\Rightarrow (3 a-2 c)<0$ .....(i)
$\text { Also, } (3 a-c)>0$ ....(ii)
$\text { So, } (3 a-2 c)(3 a-c)<0$
$\Rightarrow \left(9 a c-9 a^2-2 c^2\right)>0$