The area of the triangle formed by the tangent and the normal to the parabola `y^2=4a x ,` both drawn at the same end of the latus rectum, and the axis of the parabola is (a) `2sqrt(2)a^2` (b) `2a^2` `4a^2` (d) none of these

Options

A. `2sqrt(2)a^(2)`
B. `2a^(2)`
C. `4a^(2)`
D. none of these

Answer

C

Explanation

One end of the latus rectum is P(a,2a).
The equation of tangent PT at P(a,2a) is
`2ya=2a(x+a)ory=x+a`
The equation of normal PN at (a,2a) is
`y+x=2a+aory+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)`.