Question

Show that the $lx+my+n=0$ is a normal to the parabola $y^2=4ax$ if $a{l}^3+2al{m}^2+m^{2}n=0$.

Answer 1

Given,

$lx+my+n=0$........(1)

is eqn of normal to parabola.

Now, $y^2=4ax$..... given parabola,

Equation of normal to above parabola is

$y^2+tx=2at+a{t}^3$.......(2)

comparing eqn (1) & (2)

$\frac{l}{t}=\frac{m}{1}=\frac{n}{-(2at+a{t}^3)}$

$\therefore m=\frac{n}{-(2at+a{t}^3)}$ & $m=\frac{l}{t}$

$-m(2at+a{t}^3)=n$ $\therefore t=l/m$

$-m(\frac{2al}{m}+\frac{a{l}^3}{m^3})=n$

$-(-2al+\frac{a{l}^3}{m^2})=n$

$-(2al{m}^2+a{l}^3)=n{m}^2$

$a{l}^3+2al{m}^2+m^{2}n=0$