Question

What is the equation of tangent to the parabola $y^2=4ax$ having a slope $m$.

• A. $y=mx-a{m}^2$
• B. $y=mx+\frac{a}{m}$
• C. $y=mx-\frac{a}{m}$
• D. $y=mx+a{m}^2$

Answer 1

The correct option is B

$y=mx+\frac{a}{m}$

Given,
Parabola: $y^2=4ax$

Let the tangent of the parabola be,
$y=mx+c$

$\Rightarrow (mx+c{)}^2=4ax$

$\Rightarrow m^2{x}^2+2mcx+c^2=4ax$

$\Rightarrow m^2{x}^2+x(2mc-4a)+c^2=0$

For having a single solution

$△=0$

$\Rightarrow (2mc-4a{)}^2-4m^2{c}^2=0$

$\Rightarrow 4m^2{c}^2+16a^2-16mca-4m^2{c}^2=0$

$\Rightarrow 16a^2=16mca$

$\Rightarrow a=mc$

$\Rightarrow c=\frac{a}{m}$

Putting the value of $c$ in equation of tangent, we get

$y=mx+\frac{a}{m}$
Hence, the correct answer is Option c.