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Question

Normals are drawn from the point P with slopes m1,m2,m3 to the parabola y2=4x. If the locus of P when m1,m2=α is part of the parabola itself then the value of α is.

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Solution

We know, equation of normal to y2=4ax is
y=mx2amam3

Thus, equation of normal to y2=4x is,
y=mx2mm3

Let it passes through (h,k)
k=mh2mm3

or m3+m(2h)+k=0 ...(i)

Here, m1+m2+m3=0,

m1m2+m2m3+m3m1=2h

m1m2m3=k where m1m2=α

m3=kα it must satisfy Eq. (i)

k3α3kα(2h)+k=0

k2=α2h2α2+α2

y2=α2x2α2+α3

On comparing with y2=4xα2=4 and 2α2+α3=0

α=2

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