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The question is: Let $(a,b,c)$ denote a point in $\mathbb{R}^3$ (which is not in the origin). Find the equation of the sphere which is centered at $(a,b,c)$ that passes through the origin.

I know that the equation of a sphere with $(a,b,c)$ at the center is $$(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2.$$

I'm not really sure how to approach the question. A point in the right direction would be helpful.

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4 Answers

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You want $(x,y,z)=(0,0,0)$ to satisfy the equation of the sphere. Let $x=y=z=0$ and you will get an identity for $R^{2}$.

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You can calculate the value of $R$ using $R^2$$=$$a^2$$+b^2$$+c^2$ since as the sphere passes through the origin the radius of the sphere should be equal to the distance between the origin and the point $(a,b,c)$.

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Think about what $R$ means in the context of your equation for the sphere above - hint - it also begins with the letter r. What should that be for your sphere?

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Let $(a,b,c)=p$. The desired equation is then $\|x-p\|^2=\|p\|^2$,

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