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Question

If the normals at three points P,Q,R of the parabola y2=4ax meet at (b,c) then the centriod of the triangle PQR lies on the

A
axis of the parabola
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B
directrix of the parabola
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C
tangent at the vertex
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D
director circle
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Solution

The correct option is A axis of the parabola
Equation of the normal to the parabola y2=4ax is
y=mx2amam3 it passes through (b,c)
c=mb2amam3am3+(2ab)m+c=0
Let the roots of the equation are m1,m2,m3
Then m1+m2+m3=0m1m2+m3m2+m1m3=2abam1m2m3=ca
Feet of the normals are
P(am21,2am1)
Q(am22,2am2)
R(am23,2am3)
y coordinate of the centriod of the triangle PQR is
=2a(m1+m2+m3)3=0

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