# How can a constant point have a locus? My question is regarding this and similar problems like above. So the question is asking to find the locus of the centroid of the tetrahedron. But to my understanding, the centroid of the tetrahedron is a constant point (Because now the tetrahedron is also described according to constant coordinate points and sphere which too has constant radius). So how can a constant point like these can be represented as a equation?

Presumably, the intended meaning is that the sphere is allowed to vary (though the phrasing of the problem is rather poor and should indicate this more clearly). In other words, you consider all possible points $$A,B,C$$ that are where a sphere of radius $$2k$$ which passes through the origin meets the axes. Then, you consider the locus consisting of the centroids of the tetrahedron $$OABC$$ for all possible such $$A,B,C$$.
• Why would the points $A,B,C$ be the same? Imagine taking a sphere through the origin and rotating it around the origin. As it rotates, the points where it hits the coordinate axes will change. – Eric Wofsey Aug 24 '19 at 5:16
• @AagatPokhrel: If you really had a constant point, say, $P=(a,b,c)$, then you could give its equation as $(x-a)^2+(y-b)^2+(z-c)^2=0$: a sphere centered at $P$ with radius $0$. – Blue Aug 24 '19 at 6:11