3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade None
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 plane and also planes through $A, B, C$ parallel to coordinate planes is:
$ayz + bzx + cxy = xyz$
$axy + byz + czx = xyz$
$axy + byz + czx = abc$
$bcx + acy + abz = abc$
Step-by-Step Solution
Key Concept: Eliminate the parameters from the intercept form to obtain the locus of points satisfying all plane conditions.
A plane through points $(a, 0, 0)$, $(0, b, 0)$, $(0, 0, c)$ has equation $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$. Planes parallel to the coordinate planes and passing through these points have equations $x = a$, $y = b$, $z = c$. The locus of the common point is found by eliminating the parameters $a$, $b$, $c$ from $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1$, giving $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1$.
Correct Answer: 4