Question

The common tangent to the parabola $y^2=32x$ and $x^2=108y$ intersects the coordinate axes at the 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

The correct option is D $6\sqrt{13}$

Tangent to the parabola $y^2=32x$ is of the form $y=mx+\frac{8}{m}\cdots \left(1\right)$
is also the tangent to the parabola
$x^2=108y$
$x^2=108(mx+\frac{8}{m})\Rightarrow x^{2}m-108m^{2}x-108\times 8=0$

$D=0\Rightarrow m=-\frac{2}{3}\text{and}m=0$
But $m\ne 0\Rightarrow m=-\frac{2}{3}$
Thus, equation of the tangent will be
$y=-\frac{2}{3}x-12P(-18,0)\&Q(0,-12)$
$\Rightarrow PQ=\sqrt{{18}^2+{12}^2}=6\sqrt{13}$