Curve of intersection of two spehere is

Options

A. an ellipse
B. a circle
C. a parabola
D. None of these

Answer

To determine the curve of intersection of two spheres, we can follow these steps:

### Step 1: Understand the Geometry of Spheres
A sphere in 3D space is defined by its center and radius. The equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]
We can have two spheres, \(S_1\) and \(S_2\), with their respective equations.

### Step 2: Set Up the Equations
Let’s assume the equations of the two spheres are:
- Sphere \(S_1\): \((x - h_1)^2 + (y - k_1)^2 + (z - l_1)^2 = r_1^2\)
- Sphere \(S_2\): \((x - h_2)^2 + (y - k_2)^2 + (z - l_2)^2 = r_2^2\)

### Step 3: Find the Intersection
To find the intersection of the two spheres, we can subtract one equation from the other. This will eliminate the quadratic terms and lead us to a linear equation in \(x\), \(y\), and \(z\):
\[ (x - h_1)^2 + (y - k_1)^2 + (z - l_1)^2 - (x - h_2)^2 - (y - k_2)^2 - (z - l_2)^2 = r_1^2 - r_2^2 \]

### Step 4: Analyze the Resulting Equation
The resulting equation will represent a plane in 3D space. The intersection of this plane with either of the spheres will yield a curve.

### Step 5: Identify the Type of Curve
The intersection of a plane with a sphere is a circle. Therefore, the curve of intersection of two spheres is a circle.

### Conclusion
Thus, the curve of intersection of two spheres is a **circle**.

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