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`

Options

A. 1
B. 2
C. 5
D. 3

Answer

B

Explanation

Let the three points on the parabola be `A (at_(1)^(2), 2at_(1)), B(at_(2)^(2), 2at_(2))` and `C(at_(3)^(2), 2at_(3))`.
Equation of the tangent to the parabola at `(at^(2), 2at)` is `ty = x + at^(2)`
Therefore, equation of tangents at A and B are
`t_(1) y = x + at_(1)^(2)`
and `t_(2) y = x + at_(2)^(2)`
From Eqs (i) and (ii)
`t_(1) y = t_(2) y - at_(2)^(2 - at_(1)^(2)`
`implies y = a (t_(1) + t_(2)) [:' t_(1) =! t_(2)]`
and `t_(1) a (t_(1) + t_(2)) = x + at_(1)^(2)` [from Eq. (i))
`implies x = at_(1) t_(2)`
Therefore, coordinates of P are `(at_(1) t_(2) a (t_(1) + t_(2))`
Similarly the coordinates of Q and R are respectively,
`[at_(2) t_(3), a (t_(2) + t_(3))]` and `[at_(2) t_(2), a (t_(1) + t_(3))]`.
Let `Delta_(1)` = Area of the `Delta ABC`
`= (1)/(2) ||{:(at_(1)^(2), 2at_(1), 1),(at_(2)^(2), 2at_(2), 1),(at_(3)^(2), 2at_(2), 1):}||`
Applying `R_(3) to R_(3) - R_(2)` and `R_(2) to R_(2) - R_(1)`
`Delta_(1) = (1)/(2) ||{:(at_(1)^(2), 2at_(1), 1),(a(t_(2)^(2) - t_(1)^(1)),2a(t_(2) - t_(1)),0),(a(t_(3)^(2) - t_(2)^(2)), 2a(t_(3) - t_(2)),0):}||`
`= (1)/(2) ||{:(a(t_(2)^(2) - t_(1)^(2)),2a(t_(2) - t_(1))),(a(t_(3)^(2) - t_(2)^(2)),2a(t_(3) - t_(2))):}||`
`= (1)/(2) .a. 2a ||{:((t_(2) - t_(1)),(t_(2) + t_(1)),(t_(2) - t_(1))),((t_(3) - t_(2)),(t_(3) - t_(2))):}||`
`= a^(2) (t_(2) - t_(1)) (t_(3) - t_(2)) ||{:(t_(2) +, t_(1),1),(t_(3) +,t_(2),1):}||`
`a^(2) |(t_(2) - t_(1)) (t_(3) - t_(2)) (t_(2) - t_(3))|`
Again let `Delta_(2) =` area of the `Delta PQR`
`= (1)/(2) ||{:(at_(1)t_(2),a(t_(1) + t_(2)),1),(at_(2)t_(3),a(t_(2) + t_(1)),1),(at_(3)t_(1),a(t_(3)+ t_(1)),1):}||`
`= (1)/(2) a.a ||{:(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)),1):}||`
Applying `R_(3) to R_(3) - R_(2), R_(2) to R_(2) - R_(1)` we get
`= (a^(2))/(2) ||{:(t_(1)t_(2),t_(1) + t_(2),1),(t_(2)(t_(3) - t_(1)),t_(3) - t_(1),0),(t_(3)(t_(1)-t_(2)),t_(1) - t_(2),0):}||`
`= (a^(2))/(2) (t_(3) - t_(1)) (t_(1) - t_(2)) ||{:(t_(1)t_(2),t_(1) + t_(2),1),(t_(2),1,0),(t_(3),1,0):}||`
`= (a^(2))/(2) (t_(3) - t_(1))(t_(1) t_(2)) ||{:(t_(2),1),(t_(3),1):}||`
`= (a^(2))/(2) | (t_(3) - t_(1)) (t_(1) - t_(2)) (t_(2) - t_(3))|`
Therefore, `(Delta_(1))/(Delta_(2)) = (a^(2) | (t_(2) - t_(1)) (t_(3) - t_(2)) (t_(1) - t_(3))|)/((1)/(2) a^(2) | (t_(3) - t_(1))(t_(1) - t_(2)) (t_(2) - t_(3))|)`