How to find the coordinates of one point on the interaction of the sphere $$(x-1)^2+(y-2)^2+(z-4)^2= 25$$ and the plane $z=4$.
I was trying to solve this I got it down to $x+y=8$ but then when I tried to solve it did not work. Please help me.
2026-03-31 12:16:36.1774959396
finding the coordinates of a point of intersection: 3d sphere and plane
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Okay, so you first plug in z=4 (goes to 0):
$(x-1)^2+(y-2)^2+=25$
You should recognize this as a circle centered at $(1,2)$ with radius $5$. Based on this, you can find different points. For example, you can have $(1,7,4)$ as a solution. If you got down to $x+y=8$, you went a little farther than necessary. The trick is to notice the circle. However, if you notice, I have 1+7=8