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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
answered Jun 18, 2013 by thanvigandhi_1

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