Consider the parabola `y^2=4xdot` Let `A-=(4,-4)` and `B-=(9,6)` be two fixed points on the parabola. Let `C` be a moving point on the parabola between `Aa n dB` such that the area of the triangle `A B C` is maximum. Then the coordinates of `C` are (a) `(1/4,1)` (b) `(4,4)` (c) `(3,2/(sqrt(3)))` (d) `(3,-2sqrt(3))`

Options

A. (1/4,1)
B. (4,4)
C. `(3,2sqrt(3))`
D. `(3,-2sqrt(3))`

Answer

A

Explanation

(1) The area of triangle ABC is maximum if CD is maximum, because AB is fixed.
Therefore, the tangent drawn to the parabola at C should be parallel to AB.
Slope of AB `=(6+4)/(9-4)=2`
For `y^(2)=4x`,
`(dy)/(dx)=(2)/(y)=2`
`ory=1`
`orx=(1)/(4)`