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 plane through A, B, C parallel to coordinate planes is
Answer & explanation
Correct answer: option 1
Let the equation to the plane be $\frac{x}{\alpha}+\frac{y}{\beta}+\frac{z}{\gamma}=1$
$\Rightarrow \frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}=1$ (∵ the plane passes through a, b, c)
Now the points of intersection of the plane with the coordinate axes are $A(\alpha, 0,0)$, $B(0, \beta, 0)$ & $C(0,0, \gamma)$
⇒ Equation to planes parallel to the coordinate planes and passing through A, B & C are $x=\alpha, y=\beta$ and $z=\gamma$.
∴ The locus of the common point is
$\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1$ (by eliminating $\alpha, \beta, \gamma$ from above equation)
Hence (1) is the correct answer.