The Ellipse Story

Given that, the distance between smaller circle and ellipse is 6 .

So, the length of the rectangle is

2*6= 12

Given, area of the rectangle= 48

12*breadth =48

Breadth= 4.

Let the equation of bigger ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Given that, the area of the rectangle is maximum.

We know that, if the area of the rectangle is maximum in an ellipse ,

Then ,length= a $\sqrt{2}$

a= $\frac{12}{\sqrt{2}}$ ; $a^2=$ 72

Breadth= b $\sqrt{2}$

b= $\frac{4}{\sqrt{2}}$ . ; $b^2=$ 8

Now, equation of tangent ,

For ellipse $\implies y=mx+\sqrt{a^{2}m^{2}+b^{2}}$

For circle $\implies y=mx+r(\sqrt{m^{2}+1})$

Since, it is a common tangent, comparing the constant ,

$\sqrt{a^{2}m^{2}+b^{2}}=r(\sqrt{m^{2}+1})$

$a^{2}m^{2}+b^{2}=r^2(m^{2}+1)$

Given, r= 6

$72m^{2}+8=36(m^{2}+1)$

On simplifying,

$m^2=\boxed{\frac{7}{9}}$