Question

Area of quadrilateral formed by common tangents to ellipses $E_1:\frac{x^2}{16}+\frac{y^2}{9}=1$ and $E_2:\frac{x^2}{9}+\frac{y^2}{16}=1$

• A. $25$
• B. $50$
• C. $100$
• D. $\frac{49}{2}$

Answer 1

The correct option is B $50$

Equation of tangent to the ellipse $\frac{x^2}{a^2}+\frac{x^2}{b^2}=1$

is $y=mx\pm \sqrt{a^2{m}^2+b^2}$

Now, tangent on $E_1:y=mx\pm \sqrt{16m^2+9}{E}_2:y=mx\pm \sqrt{9m^2+16}$

For Common Tangent $:\sqrt{9m^2+16}=\sqrt{16m^2+9}\Rightarrow m^2=1\Rightarrow m=\pm 1$

So, Equation of common Tangents are

$y=x\pm 5y=-x\pm 5$

which are represented in the Graph forming a square of side $=\sqrt{5^2+5^2}=5\sqrt{2}$

Hence, Area of the square is $5\sqrt{2}\times 5\sqrt{2}=50$