If `P` is a point on the parabola `y^2=3(2x-3)` and `M` is the foot perpendicular drawn from `P` on the directrix of the parabola, then the length of each side of the equilateral triangle `S M P ,` wher `S` is the focus of the parabola, is 2 (b) 4 (c) 6 (d) 8

Options

A. 2
B. 4
C. 6
D. 8

Answer

C

Explanation

(3) `y^(2)=6(x-(3)/(2))`
The equation of directrix is
`x-(3)/(2)=-(3)/(2)orx=0`
Vertex is `((3)/(2),0)` and focus of P be `((3)/(2)+(3)/(2)t^(2),3t)`.
Then the coordinates of M are (0,3t). Therefore,
`MS=sqrt(9+9t^(2))`
`MP=(3)/(2)+(3)/(2)t^(2)`
`or9+9t^(2)=((3)/(2)+(3)/(2)t^(2))^(2)=(9)/(4)(1+t^(2))^(2)`
`or4=1+t^(2)`
`:.` Length of side = 6