Question

The common tangent to the parabola $y^{2}=32 x$ and $x^{2}=256 y$ will be_________

• A. $2 x+4 y+64=0$
• B. $x+2 y-32=0$
• C. $2 x+4 y+32=0$
• D. $4 x x+2 y+64=0$

Answer 1

Answer: (A)

Given parabolas are,
$y^{2}=32 x$ and $x^{2}=256 y$
We use a standard result to find equation of common tangent.
Equation of tangent common to $y^{2}=4 a x$ and $x^{2}=4 b y$ is,
$b^{1 / 3} y +a^{1 / 3} x+\left(a^{2} b^{2}\right)^{1 / 3}=0$
Here, $4 a=32 \Rightarrow a=8$
$4 b=256, b=64$
So tangent is,
$(64)^{1 / 3} y+(8)^{1 / 3} x+\left(8^{2} \times 64^{2}\right)^{1 / 3}=0$
$\Rightarrow 4 y+2 x+64=0$