Final answer:
The equation of the sphere touching the XY-plane is (x - 2)² + (y - 3)² + (z - 5)² = 25. The equation of the sphere touching the YZ-plane is (x - 2)² + (y - 3)² + (z - 5)² = 4. The equation of the sphere touching the XZ-plane is (x - 2)² + (y - 3)² + (z - 5)² = 9.
Step-by-step explanation:
To find the equations of the spheres with center (2, 3, 5) that touch the given planes, we need to determine the radius of each sphere based on the distance from the center to the point of tangency on the plane.
Finding the Equation of a Sphere
The general equation of a sphere with center (h, k, l) and radius r is given by:
(x - h)² + (y - k)² + (z - l)² = r²
Sphere Touching the XY-plane
The center of the sphere is at (2, 3, 5). The XY-plane has equation z = 0. The distance from the center to the XY-plane is the absolute value of the z-coordinate of the center, which is 5. Therefore, the radius of the sphere is 5, and the equation of the sphere is:
(x - 2)² + (y - 3)² + (z - 5)² = 5²
Sphere Touching the YZ-plane
The YZ-plane has equation x = 0. The distance from the center to the YZ-plane is the absolute value of the x-coordinate of the center, which is 2. Thus, the radius of the sphere is 2, and the equation of the sphere is:
(x - 2)² + (y - 3)² + (z - 5)² = 2²
Sphere Touching the ZX-plane
There is a typo in the question; I assume 'Z-plane' refers to the XZ-plane or ZX-plane, which has the equation y = 0. The distance from the center to the XZ-plane is the absolute value of the y-coordinate of the center, which is 3. Thus, the radius of the sphere is 3, and the equation of the sphere is:
(x - 2)² + (y - 3)² + (z - 5)² = 3²