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 B

$128$

Explanation for the correct answer:

Find the value of $b$:

Given,

$y=mx+4$ is a tangent to both the parabolas,$y^2=4x$ and $x^2=2by$.

We know,

Any tangent to the parabola ${\mathbf{y}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4}\mathbf{ax}$ is $\mathbf{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{mx}\mathbf{}\mathbf{+}\mathbf{}\frac{\mathbf{a}}{\mathbf{m}}$.

So,

For parabola $y^2=4x$, equation of tangent must be $y=mx+\frac{1}{m}$ as value of $a=1$

From the given equation $y=mx+4$ we can say $\frac{1}{m}=4\Rightarrow m=\frac{1}{4}$

Now, equation of the tangent becomes $y=\frac{1}{4}x+4$

Now,

$y=\frac{1}{4}x+4$ is tangent to $x^2=2by$,

Then,

$\begin{array}{rcl}2b\left(\frac{1}{4}x+4\right)& =& x^2\\ & \Rightarrow & 2x^2-bx-16b=0\end{array}$

Now make discriminant equal to zero,

$$\begin{gathered} b^2-4ac=0 \\ \Rightarrow b^2-4\left(2\right)\left(-16b\right)=0 \\ \Rightarrow b^2+128b=0 \\ \Rightarrow b=-128 \end{gathered}$$

Hence, the correct option is C.