Question

The equation of the line which is tangent to both the circle $x^2+y^2=5$ and the parabola $y^2=40x$ is

• A. 2x + y + 5 = 0
• B. 2x - y - 5 = 0
• C. $2x-y\pm 5=0$
• D. 2x - y + 5 = 0

Answer 1

Answer: (C)

Let the equation of line be $y=mx+c$
Since this line is tangent to the circle
$x^2+y^2=5$
$\therefore c=\pm a\sqrt{1+m^2}$
$=\pm \sqrt{5}\sqrt{1+m^2}\dots (i)$
Also, the above line is tangent to the parabola
$y^2=40x$
$\therefore c=\frac{a}{m}=\frac{10}{m}\dots (ii)$
From Eqs. (i) and (ii)
$\Rightarrow \frac{10}{m}=\pm \sqrt{5}\sqrt{1+m^2}$
$\Rightarrow \frac{20}{m^2}=1+m^2$
$\Rightarrow m^4+m^2-20=0$
$\Rightarrow \left(m^2+5\right)\left(m^2-4\right)=0$
$\Rightarrow m^2=4,m^2\ne -5$
$\Rightarrow m=\pm 2$
$\Rightarrow c=\pm 5$
$\therefore y=\pm 2x\pm 5$