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

Answer

To find the value of \( y \) given the center of the circle and the endpoints of its diameter, we can follow these steps:

### Step 1: Identify the coordinates
The center of the circle is given as \( (3, 5) \). The endpoints of the diameter are \( (4, 7) \) and \( (2, y) \).

### Step 2: Use the midpoint formula
The center of the circle is the midpoint of the diameter. The midpoint \( M \) of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 3: Set up the equations
Let the endpoints of the diameter be:
- \( A(4, 7) \) (first endpoint)
- \( B(2, y) \) (second endpoint)

Using the midpoint formula, we can set up the equations for the x-coordinates and y-coordinates:
\[ \text{Midpoint} = \left( \frac{4 + 2}{2}, \frac{7 + y}{2} \right) = (3, 5) \]

### Step 4: Solve for x-coordinate
From the x-coordinates:
\[ \frac{4 + 2}{2} = 3 \]
This is already satisfied since \( \frac{6}{2} = 3 \).

### Step 5: Solve for y-coordinate
From the y-coordinates:
\[ \frac{7 + y}{2} = 5 \]

### Step 6: Multiply both sides by 2
To eliminate the fraction, multiply both sides by 2:
\[ 7 + y = 10 \]

### Step 7: Solve for \( y \)
Now, isolate \( y \):
\[ y = 10 - 7 \]
\[ y = 3 \]

### Final Answer
Thus, the value of \( y \) is \( 3 \).

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