Question: If the focus of a parabola divides a focal chord of the parabola in segments of length 3 and 2, the ...

Question

If the focus of a parabola divides a focal chord of the parabola in segments of length 3 and 2, the length of the latus rectum of the parabola is-

Answer 1

Solution

Let y² = 4ax be the equation of the parabola, then the focus is S(a, 0). Let P(at₁², 2at₁) and Q(at₂², at₂) be vertices of a focal chord of the parabola, then t₁t₂ = –1. Let SP = 3,

SQ = 2

SP = $\sqrt{a^{2}(1 - t_{1}^{2}) + 4a^{2}t_{1}^{2}}$ = a(1 + t₁²) = 3 (i)

and SQ = a $\left( 1 + \frac{1}{t_{1}^{2}} \right)$ = 2

From (i) and (ii) we get t₁² = 3/2 and a = 6/5

Hence the length of the latus rectum = 24/5.