Question

A ling touches the circle $x^2+y^2=2$ and the parabola $y^2=8x$, then equation of tangent is

• A. $y=x+3$
• B. $y=x+2$
• C. $y=x+4$
• D. $y=x+1$

Answer 1

The correct option is B $y=x+2$

Let the equation of line be $y=mx+c$. If this line touches the parabola $y^2=8x$, then
$y=mx+\frac{2}{m}$
This line also touches the circle $x^2+y^2=2$, then
radius $=$ perpendicular distance from centre $(0,0)$ to the line
$\Rightarrow \sqrt{2}=\left|\frac{0+0-\frac{2}{m}}{\sqrt{1+m^2}}\right|$
$\Rightarrow 2=\frac{4}{m^2(1+m^2)}$
$\Rightarrow m^2(1+m^2)=2\Rightarrow m=1$
$\therefore$ The required equation of tangent be
$y=x+2$