A parabola having directrix `x +y +2 =0` touches a line `2x +y -5 = 0` at (2,1). Then the semi-latus rectum of the parabola, is

Options

A. 8
B. `(9)/(sqrt(2))`
C. `(10)/(sqrt(2))`
D. `(11)/(sqrt(2))`

Answer

B

Explanation

Let foot of perpendicular from (2,1) on the dirextrix be `(alpha,beta)`
`:. (alpha-2)/(1) = (beta-1)/(1) = (-(2+1+2))/(2)`
`rArr alpha =- (1)/(2), beta =- (3)/(2)`
Now, image of point `(alpha, beta)` in the tangent is focus which is `(gamma, delta)`.
`:. (gamma+(1)/(2))/(2)=(delta+(3)/(2))/(1) =(-2(-1-(3)/(2)-5))/(5)=3`
`rArr S -= (gamma, delta) -= ((11)/(2),(3)/(2))`
Semi-latus rectum = Distance of focus from directrix
`=(|(11)/(2)+(3)/(2)+2|)/(sqrt(2)) =(9)/(sqrt(2))`