Fill up in each of the following
The equation of the circle which passes through the point (4 , 6) and has its centre at (1 , 2) is

Answer

To find the equation of the circle that passes through the point (4, 6) and has its center at (1, 2), we will follow these steps:

### Step 1: Identify the center and the point on the circle
The center of the circle is given as \( (h, k) = (1, 2) \) and a point on the circle is \( (x_1, y_1) = (4, 6) \).

### Step 2: Calculate the radius using the distance formula
The radius \( r \) can be calculated using the distance formula:
\[ r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} \]
Substituting in the values:
\[ r = \sqrt{(4 - 1)^2 + (6 - 2)^2} \]
Calculating the differences:
\[ = \sqrt{(3)^2 + (4)^2} \]
\[ = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \]

### Step 3: Write the general 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 \( h = 1 \), \( k = 2 \), and \( r = 5 \):
\[ (x - 1)^2 + (y - 2)^2 = 5^2 \]
\[ (x - 1)^2 + (y - 2)^2 = 25 \]

### Step 4: Expand the equation
Now, we will expand the equation:
\[ (x - 1)^2 + (y - 2)^2 = 25 \]
Expanding both squares:
\[ (x^2 - 2x + 1) + (y^2 - 4y + 4) = 25 \]
Combining like terms:
\[ x^2 + y^2 - 2x - 4y + 5 = 25 \]
Subtracting 25 from both sides:
\[ x^2 + y^2 - 2x - 4y + 5 - 25 = 0 \]
\[ x^2 + y^2 - 2x - 4y - 20 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 2x - 4y - 20 = 0 \]