Find the coordinates of a point on the y-axis which is equidistant from A(1,-3,7) and B(5,7,-5)
I understand an isosceles triangle can be formed from these points, and the x,z coordinates of this point must be 0, but I'm struggling to see how I could calculate the y-coordinate?
Trace of points which have equidistant of $A$ and $B$ is a plane perpendicular to $AB$ and passes midpoint of $AB$ i.e. $2x+5y-7z=9.$
Suppose plane is $ax+by+cz=d.$
perpendicular to $AB$: $(a,b,c)=(5-1,7-(-3),-5-7)$
passes midpoint of $AB$: $2\times 3+5 \times 2-7\times 1=d$
If you want to find your point on this plane, note that $x=z=0 \Rightarrow y=9/5.$