The equation of the sphere on the join of `(2,3,5),(4,9,-3)` as diameter is …......

Answer

To find the equation of the sphere with the endpoints of the diameter at the points \( (2, 3, 5) \) and \( (4, 9, -3) \), we can follow these steps:

### Step 1: Find the center of the sphere
The center of the sphere is the midpoint of the line segment joining the two endpoints of the diameter. The midpoint \( M(x, y, z) \) can be calculated using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting the coordinates of the points:
\[ M = \left( \frac{2 + 4}{2}, \frac{3 + 9}{2}, \frac{5 - 3}{2} \right) = \left( \frac{6}{2}, \frac{12}{2}, \frac{2}{2} \right) = (3, 6, 1) \]

### Step 2: Find the radius of the sphere
The radius \( r \) of the sphere is half the distance between the two endpoints. The distance \( d \) between the points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Calculating the distance:
\[ d = \sqrt{(4 - 2)^2 + (9 - 3)^2 + (-3 - 5)^2} = \sqrt{(2)^2 + (6)^2 + (-8)^2} = \sqrt{4 + 36 + 64} = \sqrt{104} \]
Thus, the radius \( r \) is:
\[ r = \frac{d}{2} = \frac{\sqrt{104}}{2} = \frac{2\sqrt{26}}{2} = \sqrt{26} \]

### Step 3: Write the equation of the sphere
The standard equation of a sphere with center \( (h, k, l) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]
Substituting \( (h, k, l) = (3, 6, 1) \) and \( r = \sqrt{26} \):
\[ (x - 3)^2 + (y - 6)^2 + (z - 1)^2 = 26 \]

### Step 4: Expand the equation
Expanding the equation:
\[ (x - 3)^2 = x^2 - 6x + 9 \]
\[ (y - 6)^2 = y^2 - 12y + 36 \]
\[ (z - 1)^2 = z^2 - 2z + 1 \]
Combining these, we have:
\[ x^2 - 6x + 9 + y^2 - 12y + 36 + z^2 - 2z + 1 = 26 \]
Simplifying:
\[ x^2 + y^2 + z^2 - 6x - 12y - 2z + 46 = 26 \]
\[ x^2 + y^2 + z^2 - 6x - 12y - 2z + 20 = 0 \]

### Final Answer
The equation of the sphere is:
\[ x^2 + y^2 + z^2 - 6x - 12y - 2z + 20 = 0 \]