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# Find the centre and radius of the following spheres: $\overrightarrow{r}^{2}-\overrightarrow{r} . (\overrightarrow{4i}+\overrightarrow{2j}-\overrightarrow{6k})-11=0$

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• General equation of a sphere $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ centre $(-u, -v, -w)$ radius $= \sqrt{u^2+v^2+w^2-d}$
$\overrightarrow r^2-\overrightarrow r.(4\overrightarrow i+2\overrightarrow j-6\overrightarrow k)-11=0$
Let $\overrightarrow r=x\overrightarrow i+y\overrightarrow j+z\overrightarrow k$
$\therefore \overrightarrow r^2=x^2+y^2+z^2\: and \: \overrightarrow r.(4\overrightarrow i+2\overrightarrow j-6\overrightarrow k)=4x+2y-6z$
The equation of the sphere is $x^2+y^2+z^2+4x+2y-6z-11=0$. The centre is at $( -u, -v, -w)$ where $2u=4, \: 2v=2, \: 2w=-6$
$\therefore c(2, 1, -3)\: and \: r=\sqrt{u^2+v^2+w^2-d}$ where $d=-11$
The radius is $\sqrt{4+1+9+11}=\sqrt{25}=5$ units