Question

If $y=4x+3$ is parallel to a tangent to the parabola $y^2=12x$, then its distance from the normal parallel to the given line is

• A. $\frac{213}{\sqrt{17}}$
• B. $\frac{219}{\sqrt{17}}$
• C. $\frac{211}{\sqrt{17}}$
• D. $\frac{210}{\sqrt{17}}$

Answer 1

The correct option is B $\frac{219}{\sqrt{17}}$

Equation of normal to $y^2=4ax$ having slope $m$ is

$y=mx-2am-a{m}^3$

So, the equation of normal to $y^2=12x$ parallel to $y=4x+3$ is

$y=4x-2\left(3\right)\left(4\right)-\left(3\right)(4{)}^3$

$⟹y=4x-216$

Distance between the lines $y=4x-216$ and $y=4x+3$

$=\left|\frac{3-(-216)}{(4{)}^2+(-1{)}^2}\right|$

$=\frac{219}{\sqrt{17}}$

So, the answer is option (B).