# Find the equation of the sphere with points P such that the distance from P to A is twice the distance from P to B. A(-2,4, 2), B(5,2, -1)

Find the equation of the sphere with points P such that the distance from P to A is twice the distance from P to B.
A(-2,4, 2), B(5,2, -1)
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Wayne Everett
$r=|AB|=\sqrt{\left(5+2{\right)}^{2}+\left(2-4{\right)}^{2}+\left(-1-2{\right)}^{2}}=\sqrt{\left(7{\right)}^{2}+\left(2{\right)}^{2}+\left(3{\right)}^{2}}=\sqrt{62}$
$2|PA|=|PB|$
$2\sqrt{\left(x+2{\right)}^{2}+\left(x-4{\right)}^{2}+\left(z-2{\right)}^{2}}=\sqrt{\left(x-25{\right)}^{2}+\left(x-2{\right)}^{2}+\left(z+1{\right)}^{2}}$
$4\left(\left(x+2{\right)}^{2}+\left(x-4{\right)}^{2}+\left(z-2{\right)}^{2}\right)=\left(x-5{\right)}^{2}+\left(x-2{\right)}^{2}+\left(z+1{\right)}^{2}$
simplify obtain sphere