If three parabols touch all the lines `x = 0, y = 0` and `x +y =2`, then maximum area of the triangle formed by joining their foci is

Options

A. A. `sqrt(3)`
B. B. `sqrt(6)`
C. C. `(3sqrt(3))/(4)`
D. D. `(3sqrt(3))/(2)`

Answer

D

Explanation

Let ABC be a triangle whose sides are `x =0, y =0` and `x +y = 2`. Since, the parabols touch the side, so foci must lie on circumcircle of the triangle ABC whose radius is `sqrt(2)`. Now, foci form a triangle of maximum area.
Hence, foci must be the vertices of an equilateral triangle inscribed in the circumcircle.
So, area `= (sqrt(3))/(4) (sqrt(3)sqrt(2))^(2) = (3sqrt(3))/(2)`.