Equation of circle centred on Y-axis , and passing through
(4,6) and (6,11), is

Options

A. `x^(2) + y^(2) - 21y + 74 = 0 `
B. `x^(2) + y^(2) - 21x + 74 = 0 `
C. `x^(2) + y^(2) + 21y - 74 = 0 `
D. `x^(2) + y^(2) - 21y = 0 `

Answer

To find the equation of a circle centered on the Y-axis and passing through the points (4, 6) and (6, 11), we can follow these steps:

### Step 1: Identify the center of the circle
Since the center of the circle lies on the Y-axis, we can denote the center as \( O(0, k) \), where \( k \) is the y-coordinate we need to find.

### Step 2: Use the distance formula to find the radius
The radius \( r \) of the circle can be calculated using the distance from the center \( O(0, k) \) to any point on the circle. We will use the point \( A(4, 6) \) for this calculation.

The distance formula is given by:
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of point A and the center:
\[ r = \sqrt{(4 - 0)^2 + (6 - k)^2} = \sqrt{16 + (6 - k)^2} \]

### Step 3: Set up the equation using the second point
We also know that the circle passes through point \( B(6, 11) \). Therefore, the distance from the center \( O(0, k) \) to point B must also equal the radius \( r \):
\[ r = \sqrt{(6 - 0)^2 + (11 - k)^2} = \sqrt{36 + (11 - k)^2} \]

### Step 4: Set the two expressions for the radius equal to each other
Since both expressions represent the radius \( r \), we can set them equal:
\[ \sqrt{16 + (6 - k)^2} = \sqrt{36 + (11 - k)^2} \]

### Step 5: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ 16 + (6 - k)^2 = 36 + (11 - k)^2 \]

### Step 6: Expand both sides
Expanding both sides:
\[ 16 + (36 - 12k + k^2) = 36 + (121 - 22k + k^2) \]
This simplifies to:
\[ 52 - 12k = 157 - 22k \]

### Step 7: Solve for \( k \)
Rearranging gives:
\[ 22k - 12k = 157 - 52 \]
\[ 10k = 105 \]
\[ k = 10.5 \]

### Step 8: Find the radius \( r \)
Now that we have \( k \), we can find the radius using either point. Using point A:
\[ r = \sqrt{16 + (6 - 10.5)^2} = \sqrt{16 + (-4.5)^2} = \sqrt{16 + 20.25} = \sqrt{36.25} \]

### Step 9: Write the equation of the circle
The general equation of a circle with center \( (h, k) \) and radius \( r \) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 0 \), \( k = 10.5 \), and \( r^2 = 36.25 \):
\[ x^2 + (y - 10.5)^2 = 36.25 \]

### Step 10: Expand the equation
Expanding gives:
\[ x^2 + (y^2 - 21y + 110.25) = 36.25 \]
\[ x^2 + y^2 - 21y + 110.25 - 36.25 = 0 \]
\[ x^2 + y^2 - 21y + 74 = 0 \]

Thus, the equation of the circle is:
\[ x^2 + y^2 - 21y + 74 = 0 \]