 # Find the equation of a sphere if one of its diameters has endpoints: (-8, -6, -17) and (12, 14, 3). stratsticks57jl 2022-07-29 Answered
Find the equation of a sphere if one of its diameters has endpoints: (-8, -6, -17) and (12, 14, 3).
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The center of the sphere is the midpoint between the given points.
$\left(\frac{\left(-8\right)+\left(12\right)}{2},\frac{\left(-6\right)+\left(14\right)}{2},\frac{\left(-17\right)+\left(3\right)}{2}\right)=\left(2,4,-7\right)$
the radius is the distance from the center to either point.
$\sqrt{\left(12-2{\right)}^{2}+\left(14-4{\right)}^{2}+\left(3+7{\right)}^{2}}=\sqrt{300}$
since the equation of a sphere with center (H,K,L) and radius r is.
$\left(x-H{\right)}^{2}+\left(y-K{\right)}^{2}+\left(z-L{\right)}^{2}={r}^{2}$
then the equation is.
$\left(x-2{\right)}^{2}+\left(y-4{\right)}^{2}+\left(z+7{\right)}^{2}+300$
###### Not exactly what you’re looking for? Alonzo Odom
The diameter length is
$2R=\sqrt{\left(-8-12{\right)}^{2}+\left(-6-14{\right)}^{2}+\left(-17-3{\right)}^{2}}=20\sqrt{3}$
$R=10\sqrt{3}$
the sphere equation is:
the center of the two points also the center of the sphere:
${x}_{c}=\frac{-8+12}{2}=2,{y}_{c}=\frac{-6+14}{2}=4,{z}_{c}=\frac{-17+3}{2}=-7$
so the equation of the sphere is:
$\left(x-2{\right)}^{2}+\left(y-4{\right)}^{2}+\left(z+7{\right)}^{2}={R}^{2}=300$