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Consider a sphere passing through the origin and the points $(2,1,-1),(1,5,-4),(-2,4,-6)$
What is the radius of the sphere?
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What is the radius of the sphere?
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Verified Answer
The correct answer is:
$\sqrt{14}$
Equation of sphere passing through origin is $x^{2}+y^{2}+z^{2}+2 u x+2 v y+2 w z=0$
which passes through the points $(2,1,-1),(1,5,-4)$, and $(-2,4,-6)$
$\therefore 4 u+2 v-2 w=-6$
$2 \mathrm{u}+10 \mathrm{v}-8 \mathrm{w}=-42$
and $-4 u+8 v-12 w=-56$
Erom eqns (i),
(ii) and (iii), we get $\mathrm{u}=1, \mathrm{v}=-2$ and $\mathrm{w}=3$
$\therefore$ Radius of sphere $=\sqrt{u^{2}+v^{2}+w^{2}}$
$=\sqrt{1+4+9}=\sqrt{14}$
which passes through the points $(2,1,-1),(1,5,-4)$, and $(-2,4,-6)$
$\therefore 4 u+2 v-2 w=-6$
$2 \mathrm{u}+10 \mathrm{v}-8 \mathrm{w}=-42$
and $-4 u+8 v-12 w=-56$
Erom eqns (i),
(ii) and (iii), we get $\mathrm{u}=1, \mathrm{v}=-2$ and $\mathrm{w}=3$
$\therefore$ Radius of sphere $=\sqrt{u^{2}+v^{2}+w^{2}}$
$=\sqrt{1+4+9}=\sqrt{14}$
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