The equation of the circle passing through (4, 5) having the centre (2, 2), is

Options

A. `x^(2)+y^(2)+4x+4y-5=0`
B. `x^(2)+y^(2)-4x-4y-5=0`
C. `x^(2)+y^(2)-4x=13`
D. `x^(2)+y^(2)-4x-4y+5=0`

Explanation

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 the point through which the circle passes is \( (4, 5) \).

### Step 2: Calculate the radius of the circle
The radius \( R \) of the circle can be calculated using the distance formula between the center and the point on the circle. The distance formula is given by:

\[ R = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of the center \( (2, 2) \) and the point \( (4, 5) \):

\[ R = \sqrt{(4 - 2)^2 + (5 - 2)^2} \]
\[ R = \sqrt{(2)^2 + (3)^2} \]
\[ R = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Write the equation of the circle
The general equation of a circle with center \( (h, k) \) and radius \( R \) is given by:

\[ (x - h)^2 + (y - k)^2 = R^2 \]

Substituting \( h = 2 \), \( k = 2 \), and \( R^2 = 13 \):

\[ (x - 2)^2 + (y - 2)^2 = 13 \]

### Step 4: Expand the equation
Now, we will expand the equation:

\[ (x - 2)^2 + (y - 2)^2 = 13 \]
\[ x^2 - 4x + 4 + y^2 - 4y + 4 = 13 \]
\[ x^2 + y^2 - 4x - 4y + 8 - 13 = 0 \]
\[ x^2 + y^2 - 4x - 4y - 5 = 0 \]

### Final Equation
Thus, the equation of the circle is:

\[ x^2 + y^2 - 4x - 4y - 5 = 0 \]