To prove the center of sphere is at i)($\frac{a}{2}$,$\frac{b}{2}$,$\frac{c}{2}$). ii) radius of sphere is ($\sqrt{a^2+b^2+c^2}$)/2units.

Question

if a sphere passes through the points (0,0,0),(a,0,0),(0,b,0),(0,0,c) then show that center of sphere is at i)($\frac{a}{2}$,$\frac{b}{2}$,$\frac{c}{2}$). ii) radius of sphere is( $\sqrt{a^2+b^2+c^2}$)/2units.

Answer 1

Let's call $(x,y,z)$ the center so:

$$x^2+y^2+z^2=(x-a)^2+(y-0)^2+(z-0)^2=(x-0)^2+(y-b)^2+(z-0)^2=(x-a)^2+(y-0)^2+(z-c)^2$$

and taking each equality we get:

$$(x,y,z)=(\frac{a}{2},\frac{b}{2},\frac{c}{2})$$

and once you have the center, the radius will be the distance from the center to $(0,0,0)$, for example:

$$r^2=\left(\frac{a}{2}\right)^2+\left(\frac{b}{2}\right)^2+\left(\frac{c}{2}\right)^2 \Rightarrow r=\frac{\sqrt{a^2+b^2+c^2}}{2}$$

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