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 $3=\frac{|3m+\frac{1}{m}|}{\sqrt{1+m^2}}$



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

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

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

$\Rightarrow 3m^2=1$

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

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

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

$\therefore$ Equation of common tangent is

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

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