Question

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

• A. $\sqrt{2}y=3x+1$
• B. $\sqrt{3}y=-(x+3)$
• C. $\sqrt{3}y=x+3$
• D. $\sqrt{3}y=-(3x+1)$

Answer 1

Answer: (C)

Let the common tangent to the parabola
$y^2=4x$ be
$y=mx+\frac{1}{m}$
It should be also touch the circle
$(x-3{)}^2+y^2=9$
whose centre is (3,0) and radius $=3,$ then
$\frac{|3m+1/m|}{\sqrt{1+m^2}}=3$
$\Rightarrow 3m^2=1$
$\Rightarrow m=\pm \frac{1}{\sqrt{3}}$
But $m>0$, then equation of common tangent is
$y=\frac{1}{\sqrt{3}}\cdot x+\sqrt{3}$
or $\sqrt{3}\cdot y=x+3$