A variable plane passes through a fixed point (3,2,1) and meets the x,y,z axes at A,B,C re…

Question

A variable plane passes through a fixed point $(3,2,1)$ and meets the $x,y,z$ axes at $A,B,C$ respectively. A plane is drawn parallel to the $yz$-plane through $A$, a second plane parallel to the $zx$-plane through $B$, and a third plane parallel to the $xy$-plane through $C$. Then the locus of the point of intersection of these three planes is:

Accepted Answer

Plane $\dfrac xa+\dfrac yb+\dfrac zc=1$, and the intersection point of the three parallel planes is $(a,b,c)$. Since $(3,2,1)$ lies on the plane:
$$\frac3a+\frac2b+\frac1c=1.$$
Replacing $(a,b,c)$ by the locus variable $(x,y,z)$: $\dfrac3x+\dfrac2y+\dfrac1z=1$.

Solutions · 1