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Question

Let P be any point on the plane lx+my+nz=p and Q be a point on the line OP such that OP.OQ=p2. The locus of the point Q is

A
lx+my+nzp=x2+y2+z2
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B
lx+my+nz=p(x2+y2+z2)
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C
p(lx+my+nz)=x2+y2+z2
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D
x2+y2+z2=p2
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Solution

The correct option is B p(lx+my+nz)=x2+y2+z2
The given equation of the plane is lx+my+nz=p
Let P(α,β,γ) be a point on the plane and Q(x,y,z) be a point on OP such that OPOQ=p2
lα+mβ+nγ=p
The direction ratios of OP are α,β,γ and OQ are x,y,z.
Since O,P,Q are collinear αx=βy=γz=k .....(1)
OPOQ=α2+β2+γ2x2+y2+z2=p2
k(x2+y2+z2)=p2
lα+mβ+nγ=pk(lx+my+nz)=p
Hence, the locus of Q is p(lx+my+nz)=x2+y2+z2.

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