Question

The equation of the common tangent touching the circle $(x-3)^2 +y^2=9$ and parabola $y = 4x$ above the $x$-axis is

• A. $\sqrt{3}y=3x+1$
• B. $\sqrt{3}y=-\left(x+3\right)$
• C. $\sqrt{3}y=x+3$
• D. $\sqrt{3}y=-\left(3x+1\right)$

Answer 1

Answer: (C)

Any tangent to $y^{2}=4x$ is $y=mx+\frac{1}{m}$.

It touches the circle, if $3=\frac{\left|3m+\frac{1}{m}\right|}{\sqrt{1+m^{2}}}$



$\Rightarrow 9\left(1+m^{2}\right)=\left(3m+\frac{1}{m}\right)^{2}$

$\Rightarrow 9\left(1+m^{2}\right)m^{2}=\left(3m^{2}+1\right)^{2}$

$\Rightarrow 9m^{2}+9m^{4}=9m^{4}+1+6m^{2}$

$\Rightarrow 3m^{2}=1$

$\Rightarrow m=\pm\frac{1}{\sqrt{3}}$

For the common tangent to the above $x$-axis we take

$m=\frac{1}{\sqrt{3}}$

$\therefore $ Equation of common tangent is

$y=\frac{x}{\sqrt{3}}+\sqrt{3}$

$\Rightarrow \sqrt{3}y=x+3$