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Question

Normals are drawn from the point P with slopes m1m2=α. If P lies on the parabola y2=4x itself, then α is equal to

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Solution

Let any point on the parabola y2=4x is P(h,k).
Hence equation of normal on the given point
k=mh2mm3
m3+m(2h)+k=0
m1m2m3=k
m3=kα
(kα)3kα(2h)+k=0
k2=α2h2α2+α3
Thus, the locus of (h,k) is
y2=α2x2α2+α3
On comparing it with y2=4x, we get α2=4 and 2α2+α3=0
α=2

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