If the points A(4, 3) and B(x,5) lie on the circle with centre O(2, 3), then find the value of x:

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the value of \( x \) such that the points \( A(4, 3) \) and \( B(x, 5) \) lie on the circle with center \( O(2, 3) \), we can follow these steps:

### Step 1: Calculate the distance \( OA \)
The distance from the center \( O(2, 3) \) to the point \( A(4, 3) \) can be calculated using the distance formula:

\[ OA = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of points \( O \) and \( A \):

\[ OA = \sqrt{(4 - 2)^2 + (3 - 3)^2} \]
\[ OA = \sqrt{(2)^2 + (0)^2} \]
\[ OA = \sqrt{4} = 2 \]

### Step 2: Calculate the distance \( OB \)
Now, we calculate the distance from the center \( O(2, 3) \) to the point \( B(x, 5) \):

\[ OB = \sqrt{(x - 2)^2 + (5 - 3)^2} \]
\[ OB = \sqrt{(x - 2)^2 + (2)^2} \]
\[ OB = \sqrt{(x - 2)^2 + 4} \]

### Step 3: Set the distances equal
Since both points \( A \) and \( B \) lie on the same circle, the distances \( OA \) and \( OB \) must be equal:

\[ OA = OB \]
\[ 2 = \sqrt{(x - 2)^2 + 4} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:

\[ 2^2 = (x - 2)^2 + 4 \]
\[ 4 = (x - 2)^2 + 4 \]

### Step 5: Simplify the equation
Subtract 4 from both sides:

\[ 4 - 4 = (x - 2)^2 \]
\[ 0 = (x - 2)^2 \]

### Step 6: Solve for \( x \)
Taking the square root of both sides:

\[ x - 2 = 0 \]
\[ x = 2 \]

Thus, the value of \( x \) is \( 2 \).

### Summary
The value of \( x \) such that the points \( A(4, 3) \) and \( B(x, 5) \) lie on the circle with center \( O(2, 3) \) is:

\[ \boxed{2} \]