Question

A variable plane passes through the fixed point $(a,b,c)$ and intersects the coordinate axes at $A,B,C$. The locus of the point of intersection of the three planes through $A,B,C$ and parallel to the coordinate planes is

• A. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
• B. $\frac{x}{b_C}+\frac{y}{ca}+\frac{z}{ab}=1$
• C. $ax+by+cz=1$
• D. $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1$

Answer 1

The correct option is D $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1$

Let the equation of plane passing through $(a,b,c)$
$p(x-a)+q(y-b)+r(z-c)=0$
As it intersects coordinate axes,
$x=\frac{qb+rc+pa}{p},y=\frac{qb+rc+ap}{q},z=\frac{qb+rc+ap}{r}$
$px=qy=qz=ky$
Eliminate $p,q,r$, we get
$\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1$