Question

The area of the triangle formed by the tangent and the normal to the parabola $y^2=4ax$, both drawn at the same end of the latus rectum, and the axis of the parabola is

• A. $2\sqrt{2}{a}^2$
• B. $2a^2$
• C. $4a^2$
• D. none of these

Answer 1

Answer: (C)

One end of the latus rectum is $P(a,2a)$.
The equation of tangent $PT$ at $P(a,2a)$ is
$2ya=2a(x+a)$ or $y=x+a$
The equation of normal $PN$ at $(a,2a)$ is
$y+x=2a+a$ or $y+x=3a$
Solving $y=0$ and $y=x+a$, we get
$x=a,y=0$
Solving $y=0$ and $y+x=3a$, we get
$x=3a,y=0$
The area of the triangle with vertices $P(a,2a),T(a,0)$, and $N(3a,0)$ is $4a^2$.