Question

If $y=mx+4$ is a tangent to both the parabolas, $y^2=4x$ and $x^2=2by,$ then $b$ is equal to

• A. $-64$
• B. $128$
• C. $-128$
• D. $-32$

Answer 1

The correct option is C $-128$

Any tangent to the parabola $y^2=4ax$ is $y=mx+\frac{a}{m}$
Comparing it with $y=mx+4$, we get
$\frac{1}{m}=4$ $\Rightarrow m=\frac{1}{4}$
Equation of tangent becomes $y=\frac{x}{4}+4$

$y=\frac{x}{4}+4$ is a tangent to $x^2=2by$

$\Rightarrow x^2=2b(\frac{x}{4}+4)$
or $2x^2-bx-16b=0$
$D=0\Rightarrow b^2+128b=0$
$\Rightarrow b=0$ (not possible),
$\Rightarrow b=-128$