If the centre of a circle is (3,5) and end points of a diameter are (4,7) and (2, y), then the value of y is

Options

A. 4
B. 7
C. 3
D. -3

Answer

To find the value of \( y \) given the center of the circle and the endpoints of the diameter, we can use the midpoint formula. The midpoint of a line segment is the average of the coordinates of its endpoints.

### Step-by-Step Solution:

1. **Identify the Given Points:**
- Center of the circle \( C(3, 5) \)
- Endpoint A of the diameter \( A(4, 7) \)
- Endpoint B of the diameter \( B(2, y) \)

2. **Use the Midpoint Formula:**
The midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, the midpoint \( M \) is the center \( C(3, 5) \).

3. **Set Up the Equations:**
From the midpoint formula, we can set up the following equations:
- For the x-coordinates:
\[ \frac{x_1 + x_2}{2} = 3 \]
Substituting \( x_1 = 4 \) (from point A) and \( x_2 = 2 \) (from point B):
\[ \frac{4 + 2}{2} = 3 \quad \text{(This checks out)} \]

- For the y-coordinates:
\[ \frac{y_1 + y_2}{2} = 5 \]
Substituting \( y_1 = 7 \) (from point A) and \( y_2 = y \) (from point B):
\[ \frac{7 + y}{2} = 5 \]

4. **Solve for \( y \):**
To solve for \( y \), multiply both sides by 2:
\[ 7 + y = 10 \]
Now, isolate \( y \):
\[ y = 10 - 7 \]
\[ y = 3 \]

5. **Conclusion:**
The value of \( y \) is \( 3 \).