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 ir common with the circle $x^2+y^2=9 a^2$, then

• A

  $9 a c-9 a^2-2 c^2 < 0$
• B

  $6 a c+9 a^2-2 c^2 < 0$
• Correct answer
  C

  $9 a c-9 a^2-2 c^2>0$
• D

  $6 a c+9 a^2-2 c^2>0$

Solution

Radius $=3 a$ Length of major axis $=4 c$ Now, (Radius $) < $ (Half of the length of major axis) $$ \begin{aligned} & 3 a < 2 c \\ & 9 a^2 < 4 c^2 \\ & 9 a c-9 a^2>9 a c-4 c^2 \end{aligned} $$
$9 a c-9 a^2-2 c^2>9 a c-6 c^2$ Again $3 a < 2 c$ $\Rightarrow 9 a c < 6 c^2$ $\Rightarrow 9 a c-6 c^2 < 0$ From (i) and (ii), $9 a c-9 a^2-2 c^2>0$

Hence, the correct answer is option: 'C'.