# Prove that $\alpha(t)$ lies on a sphere.

Let $\alpha(t)$ be a regular curve. Suppose there is a point $a$ in $R^3$ such that $\alpha(t)-a$ is orthogonal to $T(t)$ for all t. Prove that $\alpha(t)$ lies on a sphere.

So, I let $\alpha(t)=(x(t), y(t),z(t))$ and $a=(a,b,c)$.

Since, $\alpha(t)-a$ is orthogonal to $T(t)$,

$<\alpha(t)-a,T(t)>=0$.

But I don't know how to solve this..

• what is $T(t)$? – Tsemo Aristide Mar 22 '18 at 13:56
• @TsemoAristide $T(t)=\alpha'(t)$, clearly. – SphericalTriangle Mar 22 '18 at 13:57
• @JooE Compute the derivative of $\|\alpha(t)-a\|^2$ – SphericalTriangle Mar 22 '18 at 13:59
• @SphericalTriangle Actually, it should be $\frac{1}{\lVert \alpha'(t)\rVert}\alpha'(t)$, because it does not state that the curve is in arclength parametrization. Not that it matters too much. – user228113 Mar 22 '18 at 14:01
• @G.Sassatelli Clearly, you don't know what you are talking about. – SphericalTriangle Mar 22 '18 at 14:07

I believe that $T(t)=\alpha'(t)$, then compute the derivative of $\|\alpha(t)-a\|^2$, it is $2<\alpha(t)-a,\alpha'(t)>$.