I have never seen a problem like this before, so I was wondering if anyone could give me help getting started. I'm studying for a quiz on Wednesday.
Find an equation of the sphere that passes through the origin and whose center is (5, 10, -9).
___ = 0
Note that you must put everything on the left hand side of the equation and that we desire the coefficients of the quadratic terms to be 1.
Similarly, how would one approach this:
Find an equation of the largest sphere with center (2, 10 , 4) that is contained completely in the first octant.
= 0
Note that you must move everything to the left hand side of the equation that we desire the coefficients of the quadratic terms to be 1.
Best Answer
Let $r$ be the radius of the sphere. Then the equation of the sphere with center $(5,10,-9)$ is $$ (x-5)^{2} + (y-10)^{2} + (z+9)^{2} =r^{2}$$
Now since your sphere passes through the origin$=(0,0,0)$, therefore you get $r^{2}=5^{2}+10^{2}+9^{2}=25+100+81=206$. Therefore the equation of the sphere is: $$(x-5)^{2}+(y-10)^{2}+(z-9)^{2}=206$$
For the next one:
If the sphere is in the 1st octant, the largest possible sphere must be tangent to either the $xy, xz$, $yz$ planes. That is, the radius of the sphere must be such that the surface of the sphere just touches one of the planes.