Question

If the common tangents of $x^{2}+y^{2}=r^{2}$ and $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ form a square, then the area (in sq. units) of the square is

• A. $50$
• B. $100$
• C. $25$
• D. $40$

Answer 1

Answer: (A)

Let, the equation of tangent in slope form for the circle be $y=m x \pm r \sqrt{1+m^{2}}$
and the equation of tangent in slope form for the ellipse be $y=m x \pm \sqrt{16 m^{2}+9}$
Now, for common tangets, $y=m x \pm r \sqrt{1+m^{2}}$ is same as $y=m x \pm \sqrt{16 m^{2}+9}$
$\Rightarrow r^{2}+r^{2} m^{2}=16 m^{2}+9 \Rightarrow \left(r^{2}-16\right) m^{2}+\left(r^{2}-9\right)=0$
Since, here $m_{1} m_{2}=-1$
$\Rightarrow 2 r^{2}=25 \Rightarrow r=\frac{5}{\sqrt{2}}$
So, the side of the square $=2 r=5 \sqrt{2}$
$\Rightarrow $ area of the square is 50 sq. units