Question

If T be any point on the tangent at any point P of a parabola, and if TL be perpendicular to the focal radius SP and TN be perpendicular to the directrix, prove that SL = TN.

Answer 1

Let the equation of parabola be $y^2=4ax$

Point $P$ be $(a{t}^2,2at)$ and the point $T$ be $(h,k)$

Equation of tangent at $P$ is

$ty=x+a{t}^2$

It passes through $T(h,k)$

$tk=h+a{t}^2......\left(i\right)$

Slope of $SP=\frac{2at-0}{a{t}^2-a}=\frac{2t}{t^2-1}$

$TL$ is perpendicular to $SP$

Then equation of $TL$ is

$2ty+(t^2-1)x-2kt-(t^2-1)h=\mathrm{0......}\left(ii\right)$

$SL=$ perpendicular distance of $S(a,0)$ from $\left(ii\right)$

$SL=\frac{|(t^2-1)x-2kt-(t^2-1)h|}{\sqrt{4t^2+{(t^2-1)}^2}}SL=\frac{|-h-h{t}^2-a-{at}^2|}{\sqrt{{(t^2+1)}^2}}SL=\frac{(a+h){(t}^2+1)}{(t^2+1)}SL=(a+h)........\left(iii\right)$

Equation of directrix is $x=-a.......\left(iv\right)$

$TN=$ perpendicular distance of $T(h,k)$ from $\left(iv\right)$

$TN=\frac{h\left(1\right)+k\left(0\right)+a}{\sqrt{1^2+0^2}}TN=h+a.......\left(v\right)$

From $\left(iii\right)$ and $\left(v\right)$

$SL=TN$

Hence proved,