Find thte area of a circle, whose centre is (5, -3) and which passes through the point (-7, 2).

Answer

To find the area of a circle with a given center and a point on the circle, we can follow these steps:

### Step 1: Identify the center and the point on the circle
The center of the circle is given as \( (5, -3) \) and a point on the circle is \( (-7, 2) \).

### Step 2: Use the distance formula to find the radius
The distance formula to find the distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, we will use the center \( (5, -3) \) as \( (x_1, y_1) \) and the point \( (-7, 2) \) as \( (x_2, y_2) \).

### Step 3: Substitute the coordinates into the distance formula
Substituting the values into the formula:
\[ d = \sqrt{((-7) - 5)^2 + (2 - (-3))^2} \]
This simplifies to:
\[ d = \sqrt{(-12)^2 + (5)^2} \]

### Step 4: Calculate the squares
Calculating the squares:
\[ d = \sqrt{144 + 25} \]

### Step 5: Add the squares and take the square root
Now, adding the squares:
\[ d = \sqrt{169} \]
Taking the square root gives:
\[ d = 13 \]
Thus, the radius \( r \) of the circle is \( 13 \).

### Step 6: Calculate the area of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius we found:
\[ A = \pi (13^2) = \pi \times 169 \]

### Step 7: Approximate the area
Using \( \pi \approx 3.14 \):
\[ A \approx 3.14 \times 169 \approx 531.46 \]
Thus, the area of the circle is approximately \( 531 \) square units.

### Final Answer
The area of the circle is approximately \( 531 \) square units.

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