Find the equation of sphere whose centre is `(5, 2, 3)` and radius is 2 in cartesian form .

Answer

To find the equation of the sphere with center at the point (5, 2, 3) and radius 2 in Cartesian form, we can follow these steps:

### Step 1: Identify the center and radius
We are given:
- Center of the sphere, \( (x_1, y_1, z_1) = (5, 2, 3) \)
- Radius, \( r = 2 \)

### Step 2: Write the general equation of a sphere
The general equation of a sphere in Cartesian coordinates is given by:
\[ (x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 = r^2 \]

### Step 3: Substitute the center and radius into the equation
Substituting the values of \( x_1, y_1, z_1 \) and \( r \) into the equation:
\[ (x - 5)^2 + (y - 2)^2 + (z - 3)^2 = 2^2 \]
This simplifies to:
\[ (x - 5)^2 + (y - 2)^2 + (z - 3)^2 = 4 \]

### Step 4: Write the final equation
Thus, the equation of the sphere in Cartesian form is:
\[ (x - 5)^2 + (y - 2)^2 + (z - 3)^2 = 4 \]

### Optional Step: Expand the equation (if required)
If you want to expand the equation using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
1. Expand \( (x - 5)^2 \) to get \( x^2 - 10x + 25 \)
2. Expand \( (y - 2)^2 \) to get \( y^2 - 4y + 4 \)
3. Expand \( (z - 3)^2 \) to get \( z^2 - 6z + 9 \)

Combining these, we have:
\[ x^2 - 10x + 25 + y^2 - 4y + 4 + z^2 - 6z + 9 = 4 \]
Simplifying this gives:
\[ x^2 + y^2 + z^2 - 10x - 4y - 6z + 38 = 4 \]
Rearranging it results in:
\[ x^2 + y^2 + z^2 - 10x - 4y - 6z + 34 = 0 \]

### Final Answer
The equation of the sphere in Cartesian form is:
\[ (x - 5)^2 + (y - 2)^2 + (z - 3)^2 = 4 \]