If (1,-2,5) and (-3,-4, 9) are the end points of a diameter of a sphere, then the radius of the sphere is

Options

A. 6 units
B. 4 units
C. 3 units
D. 2 untis

Answer

To find the radius of the sphere given the endpoints of its diameter, we can follow these steps:

### Step 1: Identify the endpoints of the diameter
The endpoints of the diameter of the sphere are given as:
- Point A: (1, -2, 5)
- Point B: (-3, -4, 9)

### Step 2: Use the distance formula to find the length of the diameter
The length of the diameter can be calculated using the distance formula in three-dimensional space, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of points A and B into the formula:
- \(x_1 = 1\), \(y_1 = -2\), \(z_1 = 5\)
- \(x_2 = -3\), \(y_2 = -4\), \(z_2 = 9\)

Calculating the differences:
- \(x_2 - x_1 = -3 - 1 = -4\)
- \(y_2 - y_1 = -4 - (-2) = -4 + 2 = -2\)
- \(z_2 - z_1 = 9 - 5 = 4\)

Now substituting these values into the distance formula:
\[ d = \sqrt{(-4)^2 + (-2)^2 + (4)^2} \]
Calculating each term:
- \((-4)^2 = 16\)
- \((-2)^2 = 4\)
- \(4^2 = 16\)

Now summing these values:
\[ d = \sqrt{16 + 4 + 16} = \sqrt{36} \]
Thus, the length of the diameter \(d\) is:
\[ d = 6 \text{ units} \]

### Step 3: Calculate the radius of the sphere
The radius \(r\) of the sphere is half of the diameter:
\[ r = \frac{d}{2} = \frac{6}{2} = 3 \text{ units} \]

### Final Answer
The radius of the sphere is \(3\) units.

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