Question

The common tangent to the parabolas $y^2 = 32x$ and $x^2 = 108y$ intersects the coordinate axes at points P and Q respectively. Then length of PQ is

• A. $2 \sqrt{13}$
• B. $3 \sqrt{13}$
• C. $5 \sqrt{13}$
• D. $6 \sqrt{13}$

Answer 1

Answer: (D)

Equation of tangent of parabola
$y^2 = 32 x$ having slope $m$ is $y = mx + \frac{8}{m} \quad ...(i)$
Equation of tangent of parabola
$x^2 = 108 \,y$ having slope $m$ is $y = mx − 27 \,m^2\quad … (ii)$
Eqs. $(i)$ and $(ii)$ are identical
$\therefore \frac{8}{m} = -27 m^{2} $
$\Rightarrow m =-\frac{2}{3} $
Equation of common tangent of parabola is
$ y = \frac{-2}{3}x -12$
$\Rightarrow 2x +3y +36 = 0$
$\Rightarrow 2x+3y +36 = 0 $
$ \therefore P\left(-18, 0\right)$ and $Q\left(0, -12\right) $
$PQ = \sqrt{\left(18\right)^{2} +\left(12\right)^{2}} $
$= \sqrt{468}$
$ = 6\sqrt{13}$