Question

The radius of the sphere passing through the point $(\alpha ,\beta ,\gamma )$ and the circle $x^2+y^2=a^2,z=0$ is

• A. $k$
• B. $\sqrt{\frac{k^2}{4}+a^2}$
• C. $\sqrt{\frac{k^2}{4}-a^2}$
• D. $\sqrt{k^2+a^2}$

Answer 1

Answer: (B)

$\sqrt{\frac{k^2}{4}+a^2}$

Equation of any sphere passing through the given circle is

$x^2+y^2+z^2-a^2+\lambda z=0$ ...(1)

If it passes through the point $(\alpha ,\beta ,\gamma )$, then

${\alpha }^2+{\beta }^2+{\gamma }^2-a^2+\lambda y=0\Rightarrow \lambda =\frac{(a^2-{\alpha }^2-{\beta }^2-{\gamma }^2)}{\gamma }=k$

Center of the sphere (1) is $(0,0,-\frac{\lambda }{2})$ and the radius $=\sqrt{\frac{k^2}{4}+a^2}$