in the parabola y2=4ax,the locus of middle points of all chords of constant length c is - Maths - Conic Sections

Question

in the parabola y2=4ax,the locus of middle points of all
chords of constant length c is

Varun.Rawat · Accepted Answer

Let AB be the chord such that AB = c
Let the coordinates of A and B are : Aat12, 2at1 and Bat22, 2at2
If Ph,k be the middle point of AB, thenh,k = at12 + at222, 2at1 + 2at22⇒h = at12 + at222 and k = at1 + at2⇒2ha = t12 + t22   and ka= t1 + t2Now, 2ha = t12 + t22 ⇒2ha = t1+t22 - 2t1t2⇒2ha =k2a2 - 2t1t2⇒ 2t1t2 = k2a2 - 2ha⇒t1t2 = k2-2ah2a2  and  t1+t2 = kaSince, AB = c⇒AB2 = c2at12-at222 + 2at1 - 2at22 = c2⇒a2t1-t22t1+t22+4 = c2⇒a2t1+t22-4t1t2t1+t22+4 = c2⇒a2k2a2 - 4k2-2ah2a2k2a2 + 4 = c2⇒a2k2-2k2+4aha2k2+4a2a2 = c2⇒4ah-k2k2+4a2 = a2c2Hence, locus of mid point of AB is4ax-y2y2+4a2 = a2c2