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