The locus of centroid of triangle formed by a tangent to the parabola `y^(2) = 36x` with coordinate axes is (a) `y^(2) =- 9x` (b) `y^(2) +3x = 0` (c) `y^(2) = 3x` (d) `y^(2) = 9x`

Options

A. `y^(2) =- 9x`
B. `y^(2) +3x = 0`
C. `y^(2) = 3x`
D. `y^(2) = 9x`

Answer

B

Explanation

Equation of tangent to `y^(2) = 36x` at `(x_(1),y_(1))` is
`y y_(1) = 18 (x+x_(1))`
It meets axis at `A(-x_(1),0)` and `B (0,(18x_(1))/(y_(1)))`.
`:.` Centroid of `DeltaOAB, (x,y) = ((-x_(1))/(3),(6x_(1))/(y_(1)))`
`:. x_(1) =- 3x` and `y_(1) = (6x_(1))/(y)`
Since, `(x_(1),y_(1))` lies on parabola, we have
`36x_(1) = (36x_(1)^(2))/(y^(2))`
`:. y^(2) = x_(1) rArr y^(2) =- 3x`