Question

Equation(s) of the common tangent to the parabola $y^2=24x$ and the circle $x^2+y^2=18$ is/are

• A. $2x+y+3=0$
• B. $x-y+6=0$
• C. $x+y+6=0$
• D. $x-2y+8=0$

Answer 1

Answer: (B), (C)

The correct options are
B $x-y+6=0$
C $x+y+6=0$

Tangent equation to parabola $y^2=24x$ is
$y=mx+\frac{6}{m}\dots \left(i\right)$

Tangent equation to circle $x^2+y^2=18$ is
$y=mx\pm \sqrt{18}\sqrt{1+m^2}\dots \left(ii\right)$

As equation $\left(i\right)$ and $\left(ii\right)$ are identical, so
${\left(\frac{6}{m}\right)}^2=18(1+m^2)\Rightarrow m^4+m^2=2\Rightarrow m=\pm 1$

Hence required equations of tangent are
$x+y+6=0$ and $x-y+6=0$