Spherical Coordinates — Question 5

Question 5 – Solution

Strategy Recognize $\rho\sin\phi\sin\theta=y$, multiply the equation by $\rho$, and complete a square.

Conversion $$\rho^2=2\rho\sin\phi\sin\theta=2y.$$ Thus $$x^2+y^2+z^2=2y \quad\Longleftrightarrow\quad \boxed{x^2+(y-1)^2+z^2=1}.$$

Geometry This is a sphere of radius $1$ centered at $\boxed{(0,1,0)}$, so its center lies on the positive $y$-axis. The sphere is tangent to the origin.