`2x-y+4=0` is a diameter of a circle which circumscribes a rectangle ABCD. If the coordinates of A, B are (4, 6) and (1, 9) respectively, find the area of this rectangle ABCD.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Find the midpoint of line segment AB
The coordinates of points A and B are given as A(4, 6) and B(1, 9). The midpoint L of AB can be calculated using the midpoint formula:

\[ L = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting the coordinates of A and B:

\[ L = \left( \frac{4 + 1}{2}, \frac{6 + 9}{2} \right) = \left( \frac{5}{2}, \frac{15}{2} \right) \]

### Step 2: Find the slope of line segment AB
The slope \( m \) of line segment AB can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates of A and B:

\[ m = \frac{9 - 6}{1 - 4} = \frac{3}{-3} = -1 \]

### Step 3: Find the slope of line OL
Since OL is perpendicular to AB, the slope of OL will be the negative reciprocal of the slope of AB:

\[ \text{slope of OL} = 1 \]

### Step 4: Write the equation of line OL
Using the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \):

\[ y - \frac{15}{2} = 1 \left( x - \frac{5}{2} \right) \]

Simplifying this, we get:

\[ y - \frac{15}{2} = x - \frac{5}{2} \]
\[ y = x + 5 \]

### Step 5: Find the intersection of line OL and the diameter
The diameter of the circle is given by the equation \( 2x - y + 4 = 0 \). We can rewrite this as:

\[ y = 2x + 4 \]

Now, we set the equations of OL and the diameter equal to find the intersection point:

\[ x + 5 = 2x + 4 \]

Solving for \( x \):

\[ 5 - 4 = 2x - x \implies 1 = x \implies x = 1 \]

Substituting \( x = 1 \) back into the equation of OL to find \( y \):

\[ y = 1 + 5 = 6 \]

Thus, the center O of the circle is at the coordinates \( O(1, 6) \).

### Step 6: Calculate the distance OL
Now we will find the distance \( OL \) using the distance formula:

\[ OL = \sqrt{ \left( x_L - x_O \right)^2 + \left( y_L - y_O \right)^2 } \]

Substituting the coordinates of L and O:

\[ OL = \sqrt{ \left( \frac{5}{2} - 1 \right)^2 + \left( \frac{15}{2} - 6 \right)^2 } \]
\[ = \sqrt{ \left( \frac{3}{2} \right)^2 + \left( \frac{3}{2} \right)^2 } \]
\[ = \sqrt{ \frac{9}{4} + \frac{9}{4} } = \sqrt{ \frac{18}{4} } = \sqrt{ \frac{9}{2}} = \frac{3}{\sqrt{2}} \]

### Step 7: Calculate the length of rectangle ABCD
Since OL is half the length of the rectangle, the full length \( AD \) is:

\[ AD = 2 \times OL = 2 \times \frac{3}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2} \]

### Step 8: Calculate the length AB
Using the distance formula for AB:

\[ AB = \sqrt{(1 - 4)^2 + (9 - 6)^2} = \sqrt{(-3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

### Step 9: Calculate the area of rectangle ABCD
The area \( A \) of rectangle ABCD is given by:

\[ A = AB \times AD = (3\sqrt{2}) \times (3\sqrt{2}) = 9 \times 2 = 18 \]

Thus, the area of rectangle ABCD is \( \boxed{18} \).

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