Question

Equation of a common tangent to the circle $x^2+y^2=4$ and to the ellipse $2x^2+25y^2=50$ is

• A. $\sqrt{2}x+\sqrt{21}y+\sqrt{23}=0$
• B. $\sqrt{2}x-\sqrt{21}y+2\sqrt{23}=0$
• C. $\sqrt{19}x-\sqrt{2}y+2\sqrt{21}=0$
• D. $\sqrt{19}x-y+2\sqrt{20}=0$

Answer 1

Answer: (B)

Let $y=mx+c$ is common tangent of circle
$x^2+y^2=4$ and ellipse $\frac{x^2}{25}+\frac{y^2}{2}=1$
$\therefore y=mx\pm 2\sqrt{1+m^2}$
and $y=mx\pm \sqrt{25m^2+2}$ are coincide
$\because 4\left(1+m^2\right)=25m^2+2$
$21m^2=2$
$\Rightarrow m=\pm \sqrt{\frac{2}{21}}$
therefore Equation of tangent is
$y=\frac{\sqrt{2}}{\sqrt{21}}x\pm \frac{2\sqrt{23}}{\sqrt{21}}$
$\Rightarrow \sqrt{2x}-\sqrt{21}y+2\sqrt{23}=0$