Question

If a tangent having slope $2$ of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is normal to the circle $x^{2}+y^{2}+4x+1=0,$ then the value of $4a^{2}+b^{2}$ is equal to

• A. $4$
• B. $2$
• C. $16$
• D. $8$

Answer 1

Answer: (C)

Equation of tangent is $y=2x\pm\sqrt{4 a^{2} + b^{2}}$
This is normal to the circle $x^{2}+y^{2}+4x+1=0$
This tangent passes through $\left(- 2,0\right)$
$0=-4\pm\sqrt{4 a^{2} + b^{2}}\Rightarrow 4a^{2}+b^{2}=16$