Q.64. Parabola

Question 64.
A triangle ABC of area Δ is inscribed in the parabola y^2 = 4ax such that the vertex A lies at the vertex of the parabola and BC is the focal chord. Find the difference of the distances of B and C from the axis of the parabola.

Answer 64.
If B = (at^2, 2at) and C = (at'^2, 2at'),
by the property of the focal chord, tt' = - 1

Now the difference of the distances of B and C from the x-axis which is the axis of the parabola
d = l 2at - 2at' l
= 2a l t - t' l [if a > 0]
Area of traingle ABC
Δ = (1/2) modulus of determinant
l at^2, 2at l
l at'^2, 2at' l
= a^2 l t^2t' - t'^2t l
= a^2 l tt' l * l t - t' l
= a^2 l -1 l * l t - t' l
= (a/2) * 2a l t - t' l
= (a/2) d
=> d = 2Δ/a.

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