Question

Prove that the sum of the reciprocals of the segments of any
focal chord of a parabola is constant.​

Answer 1

Step-by-step explanation:

My attempt:

Let y2=4ax be a parabola. Let PQ be the focal chord through the focus S(a,0) of the parabola such that the co-ordinates of P & Q are (at21,2at1) & (at22,2at2). Then we have, t1t2=−1. Clearly, if l=PS and l′=QS, then l=(at21−a)2+(2at1−0)2−−−−−−−−−−−−−−−−−−−√=⋯=a(t21+1) and similarly, l′=a(t22+1)=at211+t21 (since, t1t2=−1).

Then 1l+1l′=⋯=1a.

Then 1l2+1l′2=1a−21ll′