Question

Find the equation of a sphere for which the circle $x^2+y^2+7y-2z+2=0,2x+3y+4z=8$ is a greatest circle.

• A. $x^2+y^2+z^2-2x+4y-6z-10=0$
• B. $x^2+y^2+z^2-2x+4y-6z+10=0$
• C. $x^2+y^2+z^2-x+2y-3z+5=0$
• D. $x^2+y^2+z^2-x+2y-3z-5=0$

Answer 1

The correct option is B $x^2+y^2+z^2-2x+4y-6z+10=0$

Equation of a sphere through given circle is
$(x^2+y^2+z^2+7y-2z+2)+\lambda (2x+3y+4z-8)=0.$
Its centre is $(-\lambda ,-\frac{7+3\lambda }{2},-\frac{4\lambda -2}{2}).$
If the given circle is a great circle, then the plane $2x+3y+4z-8=0$ passes through centre.
$\therefore -2\lambda -\frac{3}{2}(7+3\lambda )-2(4\lambda -2)-8=0.$
$\therefore (-2-\frac{9}{2}-8)\lambda -\frac{21}{2}+4-8=0,$
$\therefore \frac{29\lambda }{2}+\frac{29}{2}=0;$
$\therefore \lambda =-1.$
Putting $\lambda =-1,$ we get the equation of sphere as
$x^2+y^2+z^2-2x+4y-6z+10=0$