Question

If $M$ is the foot of the perpendicular from a point $P$ of a parabola $y^2=4ax$ to its directrix and $SPM$ is an equilateral triangle, where $S$ is the focus, then $SP$ is equal to :

• A. $a$
• B. $2a$
• C. $3a$
• D. $4a$

Answer 1

The correct option is D $4a$

Given equation of parabola is
$y^2=4ax$
Focus is $(a,0)$
Let $P(a{t}^2,2at)$ be any point on the parabola.
PM is perpendicular to directrix.
Coordinates of foot of perpendicular M on directrix will be $(-a,2at)$
Now, $PM=\sqrt{(a{t}^2+a{)}^2}$
$\Rightarrow PM=a{t}^2+a$
$PS=\sqrt{(at-a{)}^2+4a^2{t}^2}$
$\Rightarrow PS=a{t}^2+a$
$MS=\sqrt{4a^2+4a^2{t}^2}$
$\Rightarrow MS=2a\sqrt{1+t^2}$
Since, $△PMS$ is an equilateral triangle
$P{S}^2=M{S}^2$
$\Rightarrow (a{t}^2+a{)}^2=4a^2(1+t^2)$
$\Rightarrow t^4-2t^2-3=0$
$\Rightarrow t^2=3;-1$ (not possible)
$\Rightarrow t^2=3$
So, $PS=4a$