Question

A variable plane passes through a fixed point (a, b, c) and meets the coordinate axes in A, B and C. The locus of the point common to the plane through A, B and C parallel to the coordinate planes is 

(A) ayz + bzx + cxy = xyz 

(B) axy + byz + czx = xyz 

(C) axy + byz + czx = abc 

(D) bcx + acy + abz = abc

Answer 1

Answer is (A) ayz + bzx + cxy = xyz

Let the equation to the plane be

x/α + y/β + z/γ = 1 ⇒ a/α + b/β + c/γ = 1

(since the plane passes through a, b and c).

Now the points of intersection of the plane with the coordinate axes are A(α, 0, 0), B(0, β, 0) and C(0, 0, γ ).

Equations to planes parallel to the coordinate planes and passing through A, B and C are x = α, y = β and z = γ . Therefore, the locus of the common point is

a/x + b/y + c/z  = 1 (by eliminating α, β and γ from above equation)