Question

The center of the sphere having the circle $x^2+y^2+z^2-3x+4y-2z-5=0,5x-2y+4z+7=0$ as the great circle is $($ section of a sphere by a plane through its centre is known as a great circle $)$

• A. $(\frac{3}{2},-2,1)$
• B. $(1,1,1)$
• C. $(-1,-1,-1)$
• D. $(0,0,0)$

Answer 1

The correct option is C $(-1,-1,-1)$

Equation of any sphere through the given circle is

$x^2+y^2+z^2-3x+4y-2z-5+\lambda (5x-2y+4z+7)=0$

Center of the sphere is $(\frac{3-5\lambda }{2},\frac{2\lambda -4}{2},\frac{2-4\lambda }{2})$

Since the given circle is a great circle, the center of this sphere lies on the plane $5x-2y+4z+7=0$

$\Rightarrow \frac{5(3-5\lambda )}{2}-\frac{2(2\lambda -4)}{2}+\frac{4(2-4\lambda )}{2}+7=0\Rightarrow -45\lambda +45=0\Rightarrow \lambda =1$

and the coordinate of the center are $(-1,-1,-1)$