Question

If 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 , show that the locus of the point Q is p(lx + my + nz) = x² + y² + z². 

Answer 1

Let P ≡ (α, β, γ), Q ≡ (x₁ , y₁ , z₁) 

Direction ratios of OP are α, β, γ and direction ratios of OQ are x₁ , y₁ , z₁. 

Since O, Q, P are collinear, we have