Find an equation of the sphere with center that touches the xy-plane.

Question

Find an equation of the sphere with center  that touches the xy-plane.

Quizplus · Accepted Answer

The equation of a sphere with center $(a,b,c)$ and radius $r$ is $(x-a)^2+(y-b)^2+(z-c)^2=r^2$. Since the sphere in question touches the xy-plane, its center must have a z-coordinate equal to its radius. Therefore, the center of the sphere is $(11eaae29\_f487\_e4ac\_87fe\_dd4fc698725d\_TB2067\_11, 0, 11eaae29\_f487\_e4ac\_87fe\_dd4fc698725d\_TB2067\_11)$ and its radius is $11eaae29\_f487\_e4ac\_87fe\_dd4fc698725d\_TB2067\_11$. Substituting these values into the equation of a sphere, we have $(x-11eaae29\_f487\_e4ac\_87fe\_dd4fc698725d\_TB2067\_11)^2+y^2+(z-11eaae29\_f487\_e4ac\_87fe\_dd4fc698725d\_TB2067\_11)^2=(11eaae29\_f487\_e4ac\_87fe\_dd4fc698725d\_TB2067\_11)^2$. Simplifying this expression gives the equation $x^2-2\cdot11eaae29\_f487\_e4ac\_87fe\_dd4fc698725d\_TB2067\_11\cdot x+y^2+z^2-22\cdot11eaae29\_f487\_e4ac\_87fe\_dd4fc698725d\_TB2067\_11\cdot z=0$. The only answer choice that matches this equation is $\boxed{	extbf{(D)}\ 11eaae29\_f488\_32d0\_87fe\_1bc2d8d40ee1\_TB2067\_11}$.

Find an Equation of the Sphere with Center That