UPSC NDA I 2022 Mathematics Solved Paper

Concept:The centre of a sphere is found from its general equation, then the line through the origin and that centre is determined using direction ratios.Explanation:Given sphere: $x^2 + y^2 + z^2 - 2x - 3y - 4z = 0$.Compare with $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$.So $2u = -2 \Rightarrow u = -1$, $2v = -3 \Rightarrow v = -\frac{3}{2}$, $2w = -4 \Rightarrow w = -2$.Centre of sphere is $(-u, -v, -w) = (1, \frac{3}{2}, 2)$.Line passes through origin $(0,0,0)$ and centre $(1, \frac{3}{2}, 2)$.Direction ratios are $(1, \frac{3}{2}, 2)$.Equation of line: $\frac{x-0}{1-0} = \frac{y-0}{\frac{3}{2}-0} = \frac{z-0}{2-0}$.Thus $x = \frac{2y}{3} = \frac{z}{2}$.Multiply each term by the LCM: $6x = 4y = 3z$.Answer: $6x = 4y = 3z$ (Option D).