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Question

Three normals with slopes m1,m2 and m3 are drawn from a point P not on the axis of the parabola y2=4x. If m1m2=α, results in the locus of P being a part of the parabola, then the value of α is

A
1
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B
2
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C
12
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D
None of these
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Solution

The correct option is C 2
Equation of normal y=mx2amam2, passes through P(h,k)m3+(2h)m+k=0;
Let the roots be m1,m2,m3, then
m1.m2.m3=k α.m3=k
m3=kα
(kα)3+(2h)(kα)+k=0
k3+(2h)kα2kα3=0
Locus of P is y2+(2x)α2α3=0
i.e., y2=α2x2α2+α3;
If it lies on the parabola y2=4x
Then α2=4 and 2α2+α3=0
α2(α2)=0
α=2

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