Question

The equation of sphere passing through $(1,0,0),(0,1,0)$ and $(0,0,1)$ and whose centre lies on the plane $3x-y+z=2$ is

• A. $(x^2+y^2+z^2)-3(x+y+z)+1=0$
• B. $2(x^2+y^2+z^2)-3(x+y+z)+1=0$
• C. $3(x^2+y^2+z^2)-2(x+y+z)+1=0$
• D. $4(x^2+y^2+z^2)-2(x+y+z)+1=0$

Answer 1

The correct option is C $3(x^2+y^2+z^2)-2(x+y+z)+1=0$

Let the equation of plane is $x^2+y^2+z^2+2ux+2vy+2wz+d=0$
As it passes through $(1,0,0),(0,1,0)$ and $(0,0,1)$
$\Rightarrow 1+2u+d=1+2v+d=1+2w+d=0\Rightarrow u=v=w=-\frac{1}{2}(d+1)$
Centre of sphere $(-u,-v,-w)$ lies on plane $3x-y+z=2$
$\Rightarrow -3u+v-w=2;u=v=w=-\frac{1}{2}(d+1)\Rightarrow \frac{3}{2}(d+1)=2\Rightarrow d=\frac{1}{3}$
putiing these values in sphere equation we get,
$3(x^2+y^2+z^2)-2(x+y+z)+1=0$