1. **Problem:** Find an equation of the sphere with center $(-1,4,5)$ that just touches the a. xy-plane, b. yz-plane, c. xz-plane. 2. **Recall the equation of a sphere:** $$ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 $$ where $(h,k,l)$ is the center and $r$ is the radius. 3. **Important rule:** A sphere that just touches a coordinate plane is tangent to it, so the radius equals the perpendicular distance from the center to that plane. 4. **Distances from center $(-1,4,5)$ to coordinate planes:** - Distance to xy-plane ($z=0$) is $|5|=5$ - Distance to yz-plane ($x=0$) is $|-1|=1$ - Distance to xz-plane ($y=0$) is $|4|=4$ 5. **Equations of spheres tangent to each plane:** a. Tangent to xy-plane: $$ (x + 1)^2 + (y - 4)^2 + (z - 5)^2 = 5^2 = 25 $$ b. Tangent to yz-plane: $$ (x + 1)^2 + (y - 4)^2 + (z - 5)^2 = 1^2 = 1 $$ c. Tangent to xz-plane: $$ (x + 1)^2 + (y - 4)^2 + (z - 5)^2 = 4^2 = 16 $$ 6. **Explanation:** The radius is the shortest distance from the center to the plane, so the sphere just touches the plane at exactly one point. **Final answers:** - a. $$ (x + 1)^2 + (y - 4)^2 + (z - 5)^2 = 25 $$ - b. $$ (x + 1)^2 + (y - 4)^2 + (z - 5)^2 = 1 $$ - c. $$ (x + 1)^2 + (y - 4)^2 + (z - 5)^2 = 16 $$