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Question

If the normals to the parabola y2=4ax at three points P,Q and R meet at A and S is the focus, then SPSQSR is equal to

A
(SA)2
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B
(SA)3
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C
a(SA)2
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D
a(SA)3
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Solution

The correct option is B a(SA)2
Let the points A be (h,k)

Normally y=mx2amam3 passes through (h,k)

k=mh2amam3

P=(am12,2am1),Q=(am22,2am2),R=(am32,2am3)

SP=(aam12)2+(2am1)2

=a(1+m12)2

=a(1+m12)

SPSQSR=a3(1+m12)2+(1+m22)2+(1+m32)2

=a3(1+m12+m12m22+m12m22m32

By simplification it reduces to a(ba)2+k2

So we get SPSQSR=aSA2

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