Locus of a point on a variable plane

Question

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

Ok I take a variable plane $l(x-a)+m(y-b)+n(z-x)=d$.I take a point $(h,k)$ whose locus is needed.Next what to do?

Answer 1

Let the equation of variable plane is $$\frac{x}{p}+\frac{y}{q}+\frac{z}{r}=1$$

Since it passes through fixed point $(a,b,c)$ we have:$$\frac{a}{p}+\frac{b}{q}+\frac{c}{r}=1$$

The points A, B and C are simply $(p,0,0)$ , $(0,q,0)$ and $(0,0,r)$ respectively.

So the planes parallel to coordinate planes passing though A, B and C are $x=p$ , $y=q$ and $z=r$ respectively.

Their intersection is $(p,q,r)$

I believe the answer is obvious now.