Question

Tangents and normal drawn to parabola $y^2=4ax$ at point $P(a{t}^2,2at),t\ne 0$, meet the x-axis at $T$ and $N$, respectively. If $S$ is the focus of the parabola, then

• A. $SP=ST\ne SN$
• B. $SP\ne ST=SN$
• C. $SP=ST=SN$
• D. $SP\ne ST\ne SN$

Answer 1

Answer: (C)

$SP=ST=SN$

Given eqaution of parabola is
$y^2=4ax$
Let $P(a{t}^2,2at)$ be any point on the parabola.
Equation of tangent at $P(a{t}^2,2at)$ is
$ty=x+a{t}^2$
Since tangent meets x-axis at $T$ i.e. $y=0$
$\Rightarrow x=-a{t}^2$
Coordinates of $T$ are $(-a{t}^2,0)$
Equation of normal at $P(a{t}^2,2at)$ is
$y=-tx+2at+a{t}^3$
Since the normal meets x-axis at $N$ i.e. $y=0$
$\Rightarrow x=2a+a{t}^2$
Coordinates of $N$ are $(2a+a{t}^2,0)$
Also, $S\equiv (a,0)$
Hence, $SP=a+a{t}^2,ST=a+a{t}^2$
and $SN=a+a{t}^2$

Thus, $SP=ST=SN$