Question

The locus of a point at which the line joining the points (-3, 1, 2), (1, -2, 4) subtends a right angle, is

• A. $x^2 + y^2 + z^2 + 2x + y - 6z - 3 = 0$
• B. $x^2 + y^2 + z^2 + 2x - y - 6z + 3 = 0$
• C. $x^2 + y^2 + z^2 + 2x + y - 6z + 3 = 0$
• D. $x^2 + y^2 + z^2 - 2x + y - 6z + 3 = 0$

Hint:

Perpendicular lines: Dot product of direction ratios is zero.

Answer 1

The Correct Option is C

Let the given points be A(-3, 1, 2) and B(1, -2, 4). Let P(x, y, z) be a point such that $\angle APB = 90^\circ$. The direction ratios of AP are $x+3$, $y-1$, $z-2$. The direction ratios of BP are $x-1$, $y+2$, $z-4$. Since AP and BP are perpendicular, the dot product of their direction ratios is zero: $(x+3)(x-1) + (y-1)(y+2) + (z-2)(z-4) = 0$ $x^2 + 2x - 3 + y^2 + y - 2 + z^2 - 6z + 8 = 0$ $x^2 + y^2 + z^2 + 2x + y - 6z + 3 = 0$ This is the equation of a sphere.