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

Options

A. `x =1, y =4`
B. `x=4,y=1`
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 endpoints of a diameter of a circle, and the center of the circle is at \( (2x, y) \).

### Step-by-Step Solution:

1. **Identify the Points**:
- Let point A be \( (x, 3) \).
- Let point B be \( (3, 5) \).

2. **Find the Midpoint**:
- The center of the circle, which is the midpoint of the diameter, can be calculated using 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 \( (x_2, y_2) = (3, 5) \).
- Therefore, the midpoint (center of the circle) is:
\[ \left( \frac{x + 3}{2}, \frac{3 + 5}{2} \right) = \left( \frac{x + 3}{2}, 4 \right) \]

3. **Set the Midpoint Equal to the Given Center**:
- The center of the circle is also given as \( (2x, y) \).
- Thus, we can set up the following equations from the x-coordinates and y-coordinates:
\[ \frac{x + 3}{2} = 2x \quad \text{(1)} \]
\[ 4 = y \quad \text{(2)} \]

4. **Solve for \( x \)**:
- From equation (1):
\[ \frac{x + 3}{2} = 2x \]
- Multiply both sides by 2 to eliminate the fraction:
\[ x + 3 = 4x \]
- Rearranging gives:
\[ 3 = 4x - x \implies 3 = 3x \implies x = 1 \]

5. **Solve for \( y \)**:
- From equation (2):
\[ y = 4 \]

6. **Final Values**:
- Thus, the values of \( x \) and \( y \) are:
\[ x = 1, \quad y = 4 \]

### Conclusion:
The final answer is \( (x, y) = (1, 4) \).

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