Question

$A B$ is a chord of a parabola $y^2=4 a x,(a>0)$ with vertex $A . B C$ is drawn perpendicular to $A B$ meeting the axis at $C$. The projection of $B C$ on the axis of the parabola is

• A. a unit
• B. 2 a unit
• C. 8a unit
• D. 4a unit

Answer 1

Answer: (D)

$B\left(a t^2, 2 a t\right): D\left(a t^2, 0\right) $
$ m_{A B}=\frac{2 a t}{a t^2}=\frac{2}{t}: m_{B C}=-\frac{t}{2}$
$ \left.\therefore y-2 a t=-\frac{t}{2}\left(x-a t^2\right) \text { [Equation of } B C\right]$
$ \text { for } y=0,-2 a t=-\frac{t}{2}\left(x-a t^2\right)=-\frac{t}{2} x+\frac{a t^3}{2} $
$ \Rightarrow t x=4 a t+a t^3$
$ \Rightarrow x=4 a+a t^2 $
$\therefore c\left(4 a+a t^2, 0\right) $
$\therefore \text { Projection of } B C \text { on the axis is } $
$D C=A C-A D=4 a+a t^2-a t^2=4 a$