Question

If the line $2x+3y=1$ touches the parabola $y^2=4ax$, find the length of its latus rectum.

• A. $\frac{2}{7}$
• B. $\frac{8}{9}$
• C. $\frac{7}{4}$
• D. $\frac{5}{7}$

Answer 1

The correct option is B $\frac{8}{9}$

Equation of any tangent to $y^2=4ax$ is

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

$m^{2}x-my+a=0$

Comparing it with the given tangent $2x+3y-1=0$, we find

$\frac{m^2}{2}=-\frac{m}{3}=\frac{a}{-1}$

$m=-\frac{2}{3}$

$a=\frac{m}{3}=-\frac{2}{9}$

Hence, the length of the latus rectum = $4a=\frac{8}{9}$, ignoring the negative sign for the length.