The equation of the circle which passes through the point (4, 5) and has its centre at (2, 2) is .............

Answer

To find the equation of the circle that passes through the point (4, 5) and has its center at (2, 2), we can follow these steps:

### Step 1: Identify the center and the point on the circle
The center of the circle is given as (h, k) = (2, 2) and a point on the circle is (x1, y1) = (4, 5).

### Step 2: Calculate the radius
The radius (r) of the circle can be calculated using the distance formula between the center and the point on the circle:
\[ r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} \]
Substituting the values:
\[ r = \sqrt{(4 - 2)^2 + (5 - 2)^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Write the equation of the circle
The standard form of the equation of a circle with center (h, k) and radius r is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting the values of h, k, and r:
\[ (x - 2)^2 + (y - 2)^2 = (\sqrt{13})^2 \]
This simplifies to:
\[ (x - 2)^2 + (y - 2)^2 = 13 \]

### Step 4: Expand the equation
Now, we will expand the left-hand side:
\[ (x - 2)^2 = x^2 - 4x + 4 \]
\[ (y - 2)^2 = y^2 - 4y + 4 \]
Combining these, we get:
\[ x^2 - 4x + 4 + y^2 - 4y + 4 = 13 \]
This simplifies to:
\[ x^2 + y^2 - 4x - 4y + 8 = 13 \]

### Step 5: Rearranging the equation
Now, we will bring all terms to one side:
\[ x^2 + y^2 - 4x - 4y + 8 - 13 = 0 \]
This simplifies to:
\[ x^2 + y^2 - 4x - 4y - 5 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 4x - 4y - 5 = 0 \]

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