Find the locus of the mid-points of the chord of the parabola y2 = 4ax which subtend a right angle at the vertex.
y2 = 2a(x – 4a).
Let P be t1 and Q be t2 and since PQ subtends a right angle at the vertex O(0, 0) therefore as in t1 t2 = – 4. If (h, k) be the mid-point, then 2h = (t12 + t22) and 2k = 2a(t1 + t2)
or 2h = a[(t1 + t2)2 – 2t1t2]
Locus is y2 = 2a(x – 4a).
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If x + y = k is normal to y2 = 12x, then k is
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