Given the sphere defined by the equation $x^2+y^2+z^2=9$, what is the intersection of this sphere with the yz-plane?

• A. No intersection
• B. A point at the origin
• C. A circle with equation $y^2+z^2=9$
• D. A line with equation $y+z=3$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem. The sphere is defined by the equation x^2 + y^2 + z^2 = 9, and we are tasked with finding its intersection with the yz-plane. The yz-plane is defined by x = 0.
2. Step 2: Substitute x = 0 into the sphere's equation. This eliminates the x^2 term, leaving y^2 + z^2 = 9.
3. Step 3: Recognize that the resulting equation y^2 + z^2 = 9 represents a circle in the yz-plane. The center of this circle is at the origin (0, 0) in the yz-plane, and its radius is √9 = 3.
4. Step 4: Verify the geometry. The sphere intersects the yz-plane along this circle because the sphere's radius is 3, and the yz-plane slices through the sphere symmetrically.
5. Step 5: Conclude that the intersection of the sphere with the yz-plane is a circle with the equation y^2 + z^2 = 9.