The equation of the circle with centre at `(1, 3)` and radius 3 is

Options

A. `(x-1)^(2) + (y-3)^(2) = 9`
B. `(x+1)^(2) + (y+3)^(2) = (3)^(2)`
C. `(x-3)+(y-1)^(2) = 3`
D. `(x+3)^(2) + (y+1)^(2) = (3)^(2)`

Answer

To find the equation of a circle with a given center and radius, we can use the standard formula for the equation of a circle. Here’s a step-by-step solution:

### Step 1: Identify the center and radius
The center of the circle is given as (1, 3), which means:
- \( a = 1 \) (x-coordinate of the center)
- \( b = 3 \) (y-coordinate of the center)

The radius \( r \) is given as 3.

### Step 2: Write the standard equation of a circle
The standard equation of a circle with center (a, b) and radius r is given by:
\[ (x - a)^2 + (y - b)^2 = r^2 \]

### Step 3: Substitute the values into the equation
Substituting \( a = 1 \), \( b = 3 \), and \( r = 3 \) into the equation:
\[ (x - 1)^2 + (y - 3)^2 = 3^2 \]

### Step 4: Calculate \( r^2 \)
Now calculate \( r^2 \):
\[ 3^2 = 9 \]

### Step 5: Write the final equation
Substituting \( r^2 \) back into the equation gives:
\[ (x - 1)^2 + (y - 3)^2 = 9 \]

Thus, the equation of the circle is:
\[ (x - 1)^2 + (y - 3)^2 = 9 \]

### Final Answer:
The equation of the circle with center (1, 3) and radius 3 is:
\[ (x - 1)^2 + (y - 3)^2 = 9 \]

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