Q.7. Length of the focal chord of a parabola in terms of the perpendicular distance from the vertex on it

Question 7:
If a focal chord of the parabola, y^2 = 4ax,  be at a distance d from the vertex, then its length is equal to____________?

Answer 7:
Let y^2 = 4ax be the equation of the parabola, a > 0.
If P(at^2, 2at) and Q(at'^2, 2at') be the end-points of the focal chord,
then the length of the focal chord can be proved to be
L = a(t - t')^2 [considering the property of the focal chord, tt' = -1]

Area of traingle PAQ
= (1/2) d * L ... (1)
Also, area
= (1/2) modulus of determinant
lat^2, 2at l
lat'^2, 2at' l
= a^2 (t - t') [Again noting that ltt'l = 1]

=> (1/2)d * L = a^2 (t - t')
=> (1/2)d * a(t - t')^2 = a^2 (t - t')
=> (t - t') = 2a/d
=> L
= a (t - t')^2
= a * [2a/(d]^2
= 4a^3/d^2.

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