UPSC NDA I 2015 Mathematics Paper

Concept:A plane can be expressed in intercept form as $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$, where $a$, $b$, $c$ are the $x$, $y$, $z$ intercepts, respectively.Explanation:For plane $ABC$, the intercepts on the coordinate axes are $a = 3$, $b = 6$, $c = 9$.Substitute these into the intercept form: $\frac{x}{3} + \frac{y}{6} + \frac{z}{9} = 1$.To eliminate denominators, multiply the equation by the least common multiple $18$: $18 \cdot \frac{x}{3} + 18 \cdot \frac{y}{6} + 18 \cdot \frac{z}{9} = 18$.Simplify each term: $6x + 3y + 2z = 18$.This linear equation represents the unique plane passing through the three intercept points $(3,0,0)$, $(0,6,0)$, and $(0,0,9)$.Answer:Option D: $6x + 3y + 2z = 18$.