Question
A variable plane passes through a fixed point (a, b, c) and meets the co– ordinate axes in A, B. C. Show that the locus of the point common to the planes through A. B, C parallel to the co-ordinate planes is 
Solution
Let the equation of plane be 
...(1)
where OA = α, OB = β, OC = γ ∵ plane (1) passes through (a, b, c)
...(2)
The equation of plane through A (α, 0, 0) parallel to yz-plane is
x = α ...(3)
The equation of plane through B (β, 0, 0) parallel to zx-plane is
y = β ...(4)
The equation of plane through C ( γ, 0, 0) parallel to xy-plane is
z =γ ...(5)
To eliminate α, β, γ, we put the values from (3). (4), (5) in (2) and get
which is required locus.
...(1)where OA = α, OB = β, OC = γ ∵ plane (1) passes through (a, b, c)
...(2)The equation of plane through A (α, 0, 0) parallel to yz-plane is
x = α ...(3)
The equation of plane through B (β, 0, 0) parallel to zx-plane is
y = β ...(4)
The equation of plane through C ( γ, 0, 0) parallel to xy-plane is
z =γ ...(5)
To eliminate α, β, γ, we put the values from (3). (4), (5) in (2) and get
which is required locus.
