If (x, 3) and ( 3,5) are the extremities of a diameter of a circle with centre at (2,y ) , then the values of x and y are

Options

A. x= 3, y = 1
B. x= 1, y = 4
C. x= 8 , y = 2
D. None of these

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given that the points \( (x, 3) \) and \( (3, 5) \) are the extremities of a diameter of a circle with center at \( (2, y) \).

### Step-by-Step Solution:

1. **Identify the Points**:
The extremities of the diameter are given as \( A(x, 3) \) and \( B(3, 5) \).

2. **Use the Midpoint Formula**:
The center of the circle, which is also the midpoint of the diameter, is given by the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( (x_1, y_1) = (x, 3) \) and \( (x_2, y_2) = (3, 5) \).

3. **Calculate the x-coordinate of the Midpoint**:
Set the x-coordinate of the midpoint equal to 2 (the x-coordinate of the center):
\[ 2 = \frac{x + 3}{2} \]
Multiply both sides by 2:
\[ 4 = x + 3 \]
Now, solve for \( x \):
\[ x = 4 - 3 = 1 \]

4. **Calculate the y-coordinate of the Midpoint**:
Set the y-coordinate of the midpoint equal to \( y \) (the y-coordinate of the center):
\[ y = \frac{3 + 5}{2} \]
Calculate the sum:
\[ y = \frac{8}{2} = 4 \]

5. **Final Values**:
We have found that:
\[ x = 1 \quad \text{and} \quad y = 4 \]

### Conclusion:
The values of \( x \) and \( y \) are:
\[ x = 1, \quad y = 4 \]