If the tangents at the points `Pa n dQ` on the parabola `y^2=4a x` meet at `T ,a n dS` is its focus, the prove that `S P ,S T ,a n dS Q` are in GP.

Options

A. `A.P.`
B. `G.P.`
C. `H.P.`
D. none of these

Answer

B

Explanation

Let `P(at_(1)^(2),2at_(1))" and Q(at_(2)^(2), 2at_(2))` be two points on the parabola `y^(2)=4ax.` The tangents at P and Q intersect at `T(at_(1)t_(2), a(t_(1)+t_(2)))`.
`:." "ST^(2)=(at_(1)t_(2)-a)^(2)+{a(t_(1)+t_(2))-0}^(2)`
`rArr" "ST^(2)=a^(2){(t_(1)t_(2)-1)^(2)+(t_(1)+t_(2))^(2)}`
`rArr" "ST^(2)=a^(2)(t_(1)^(2)+t_(2)^(2)+t_(1)^(2)" "t_(2)^(2)+1)`
`rArr" "ST^(2)=a^(2)(t_(1)^(2)+1)(t_(2)^(2)+1)=a(t_(1)^(2)+1)a(t_(2)^(2)+1)`
`rArr" "ST^(2)=(at_(1)^(2)+a)(at_(2)^(2)+a)`
`rArr" "ST^(2)=SP.SQ" "[{:(becauseSP=x+a=a_(1)^(2)+a),("and, SQ=x+a"=at_(2)^(2)+a):}]`