Question

The equation of the sphere which passes through the points $(1,0,0),(0,1,0),(0,0,1)$ and whose centre lie on the plane $3x-y+z=2$

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

Answer 1

The correct option is C $3(x^2+y^2+z^2)-4x-4y-4z+1=0$

Let the sphere be $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ ..(i)
Given it passes through $(1,0,0),(0,1,0),(0,0,1)$
$\Rightarrow 1+2u+d=0,1+2v+d=0,1+2w+d=0$ (ii)
Centre of (i) is $C(-u,-v,-w)=(\frac{d+1}{2},\frac{d+1}{2},\frac{d+1}{2})$
Also $C$ lies in the plane $3x-y+z=2$
$\frac{3(1+d)}{2}=2\Rightarrow d=\frac{1}{3}$

$\therefore C=(\frac{2}{3},\frac{2}{3},\frac{2}{3})$
Hence, sphere is $3(x^2+y^2+z^2)-4x-4y-4z+1=0$.