question_answer

A plane passes through a fixed point (a, b, c). The locus of the foot of the perpendicular to it from the origin is the sphere

A) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-ax-by-cz=0\]

B) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-2ax-2by-2cz=0\]

C) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-4ax-4by-4cz=0\]

D) None of these  

Correct Answer: A

Solution :

[a] Let (a, b, c) be the fixed point on the variable plane Now D. R?s of OM are \[x-0,y-0,z-0\] i.e x, y, z D.R.?s of MA are x-a, y-b, z-c. Since OM perpendicular Ma \[x(x-a)+y(y-b)+z(z-c)=0\] \[\Rightarrow {{x}^{2}}+{{y}^{2}}+{{z}^{2}}-ax-by-cx=0\]