Two mutually perpendicular tangents of the parabola `y^(2)=4ax` meet the axis at `P_(1)andP_(2)`. If S is the focal of the parabola, Then `(1)/(SP_(1))+(1)/(SP_(2))` is equal to

Options

A. `(1)/(2a)`
B. `(1)/(a)`
C. `(2)/(a)`
D. `(4)/(a)`

Answer

B

Explanation

(2) Tangents at `A(t_(1))andA(t_(2))` are respectively,
`t_(1)y=x+at_(1)^(2)`
`t_(2)y=x+at_(2)^(2)`
Tangent meet axis at `P_(1)(-at_(1)^(2),0)andP_(2)(-at_(2)^(2),0)`.
`:." "SP_(1)=a+at_(1)^(2),andSP_(2)=a+at_(2)^(2)`
Since tangents are perpendicular, `t_(1)t_(2)=-1`.
`:.(1)/(SP_(1))+(1)/(SP_(2))=(1)/(a+at_(1)^(2))+(1)/(a+at_(2)^(2))`
`=(1)/(a+at_(1)^(2))+(1)/a+(a)/(t_(1)^(2))`
`=(1)/(a+at_(1)^(2))+(t_(1)^(2))/(at_(1)^(2)+a)`
`(1)/(a)`
y-x-0