Question

If the line $lx+my+n=0$ is tangent to the parabola $y^2=4ax$, then

• A. $mn=a{l}^2$
• B. $lm=a{n}^2$
• C. $ln=a{m}^2$
• D. None of the above

Answer 1

The correct option is C $ln=a{m}^2$

Given parabola, $y^2=4ax$
$\Rightarrow 2y\frac{dy}{dx}=4a$
$\Rightarrow \frac{dy}{dx}=\frac{2a}{y}$ ..... (i)
Which is the slope of tangent.
Given, $lx+my+n=0$
is an equation of tangent of the parabola
$y^2=4ax$.
$\therefore$ Slope of tangent $=-\frac{l}{m}$ ...... (ii)
From Eqs. (i) and (ii)
$\frac{2a}{y}=-\frac{l}{m}\Rightarrow y=\frac{-2am}{l}$
$\because y^2=4ax$
$\Rightarrow \frac{4a^2{m}^2}{l^2}=4ax$
$\Rightarrow x=\frac{a{m}^2}{l^2}$
On putting the values of x and y in the following equation
$lx+my+n=0$
$l\left(\frac{a{m}^2}{l^2}\right)+m\left(\frac{-2am}{l}\right)+n=0$
$\frac{a{m}^2}{l}-\frac{2a{m}^2}{l}+n=0$
$\Rightarrow \frac{a{m}^2}{l}=n\Rightarrow a{m}^2=nl$
which is the required relation.