Question

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

Answer 1

Answer: (A)

$x^2+y^2+z^2-ax-by-cz=0$

Let $A(a,b,c)$ be the fixed point on the variable plane.

Now d.r's of $OM$ are $x-0,y-0,z-0\Rightarrow x,y,z$

d.r's of $MA$ are $x-a,y-a,z-c.$

Since $OM\perp MA$

Therefore $x(x-a)+y(y-b)+z(z-c)=0\Rightarrow x^2+y^2+z^2-ax-by-cz=0$