The equation of a sphere is `x^2+y^2+z^2-10z=0`. If one end point of a diameter of the sphere is `(-3,4,-5)`, what is the other en point?

Options

A. `(-3,-4,-5)`
B. (3,4,5)
C. (3,4,-5)
D. (-3,4,-5)

Explanation

To find the other endpoint of the diameter of the sphere given one endpoint and the equation of the sphere, we can follow these steps:

### Step 1: Rewrite the equation of the sphere
The given equation of the sphere is:
\[ x^2 + y^2 + z^2 - 10z = 0 \]

We can rearrange this equation to identify the center and radius of the sphere. To do this, we complete the square for the \( z \) term.

### Step 2: Complete the square
The term involving \( z \) is \( z^2 - 10z \). We can complete the square as follows:
\[ z^2 - 10z = (z - 5)^2 - 25 \]

Substituting this back into the sphere's equation gives:
\[ x^2 + y^2 + (z - 5)^2 - 25 = 0 \]
\[ x^2 + y^2 + (z - 5)^2 = 25 \]

### Step 3: Identify the center and radius
From the equation \( x^2 + y^2 + (z - 5)^2 = 25 \), we can see that:
- The center of the sphere is at \( (0, 0, 5) \)
- The radius of the sphere is \( 5 \) (since \( 25 = 5^2 \))

### Step 4: Use the midpoint formula
Let the given endpoint of the diameter be \( A(-3, 4, -5) \) and the other endpoint be \( B(x, y, z) \). The center of the sphere is the midpoint of the diameter, which can be calculated using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Setting the midpoint equal to the center of the sphere \( (0, 0, 5) \):
\[ \left( \frac{-3 + x}{2}, \frac{4 + y}{2}, \frac{-5 + z}{2} \right) = (0, 0, 5) \]

### Step 5: Set up equations
From the midpoint coordinates, we can set up the following equations:
1. \( \frac{-3 + x}{2} = 0 \)
2. \( \frac{4 + y}{2} = 0 \)
3. \( \frac{-5 + z}{2} = 5 \)

### Step 6: Solve for \( x, y, z \)
1. From \( \frac{-3 + x}{2} = 0 \):
\[ -3 + x = 0 \implies x = 3 \]

2. From \( \frac{4 + y}{2} = 0 \):
\[ 4 + y = 0 \implies y = -4 \]

3. From \( \frac{-5 + z}{2} = 5 \):
\[ -5 + z = 10 \implies z = 15 \]

### Step 7: Conclusion
Thus, the coordinates of the other endpoint \( B \) of the diameter are:
\[ B(3, -4, 15) \]