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…
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
$9 a c-9 a^2-2 c^2 < 0$
$6 a c+9 a^2-2 c^2 < 0$
$9 a c-9 a^2-2 c^2>0$
$6 a c+9 a^2-2 c^2>0$
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),
$9 a c-9 a^2-2 c^2>0$
