Points A, B, C lie on the parabola `y^2=4ax` The tangents to the parabola at A, B and C, taken in pair, intersect at points P, Q and R. Determine the ratio of the areas of the `triangle ABC` and `triangle PQR`

Answer

2

Explanation

2 Let the three points on the parabola `y^(2)=4ax` be `A(t_(1)),B(t_(2)) and C(t_(3))`.
Area of triangle ABC,
`Delta_(1)=(1)/(2)||{:(1,at_(1)^(2),2at_(1)),(1,at_(2)^(2),2at_(2)),(1,at_(3)^(2),2at_(3)):}||`
`=a^(2)||{:(1,t_(1),t_(1)^(2)),(1,t_(2),t_(2)^(2)),(1,t_(3),t_(3)^(2)):}||`
`=|a^(2)(t_(1)-t_(2))(t_(2)-t_(3))(t_(3)-t_(1))|`
Point of intersection of tangents at A,B and C are
`P(at_(1)t_(2),a(t_(1)+t_(2))),`
`Q(at_(2)t_(3),a(t_(2)+t_(3))),`
`R(at_(3)t_(1),a(t_(3)+t_(1)))`.
Area of triangle PQR,
`Delta_(2)=(1)/(2)||{:(1,at_(1)t_(2),a(t_(1)+t_(2))),(1,at_(2)t_(3),a(t_(2)+t_(3))),(1,at_(3)t_(1),a(t_(3)+t_(1))):}||`
`=(a^(2))/(2)||{:(1,t_(1)t_(2),t_(1)+t_(2)),(1,t_(2)t_(3),t_(2)+t_(3)),(1,t_(3)t_(1),t_(3)+t_(1)):}||`
`=(a^(2))/(2)||{:(0,(t_(1)-t_(3))t_(2),t_(1)-t_(3)),(0,(t_(2)-t_(1))t_(3),t_(2)-t_(1)),(1,t_(3)t_(1),t_(3)+t_(1)):}||`
(operating `R_(1)toR_(1)-R_(2)andR_(2) to R_(2)-R_(3))`
`=(a^(2))/(2)|(t_(1)-t_(3))(t_(2)-t_(1))(t_(2)-t_(3))|` (expanding w.r.t. `C_(1))` (2)
From (1) and (2)
`(Delta_(1))/(Delta_(2))=2`