Question

The length of perpendicular from the focus $S$ of the parabola $y^2=4ax$ on the tangent at any point $P$ on the parabola is
($O$ is the origin)

• A. $\sqrt{OS\cdot SP}$
• B. $OS\cdot SP$
• C. $OS+SP$
• D. $\frac{OS}{OP}$

Answer 1

The correct option is A $\sqrt{OS\cdot SP}$

Focus of the parabola is $S(a,0)$
equation of parabola$y^2=4ax$

let, co-ordinates of $P\equiv (a{t}^2,2at)$
Equation of tangent at $P$
$yt=x+a{t}^2$

Let perpendicular length is $SQ$ from $(a,0)$ to the tangent

$SQ=\frac{|a+a{t}^2|}{\sqrt{t^2+1}}$
$SQ=|a\sqrt{1+t^2}|$
$OS=a$
$SP=\sqrt{a^2(1-t^2{)}^2+4a^2{t}^2}$
$\Rightarrow SP=a(t^2+1)$

so, $OS\cdot SP=a\cdot a(t^2+1)=a^2(t^2+1)$
$SQ=\sqrt{OS\cdot SP}$