A rectangle ABCD is inscribed in a circle with a diameter lying along the line 3y=x+10. If A=(-6,7), B=(4,7) then the area of the rectangle is

Options

A. 80 sq. units
B. 40 sq. units
C. 160 sq. units
D. 20 sq. units

Answer

To solve the problem of finding the area of rectangle ABCD inscribed in a circle with a diameter along the line \(3y = x + 10\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates of Points A and B**:
- Given points are \(A = (-6, 7)\) and \(B = (4, 7)\).
- Since both points have the same y-coordinate, the line segment AB is parallel to the x-axis.

2. **Calculate the Length of AB**:
- The length of segment AB can be calculated using the formula:
\[ \text{Length of AB} = |x_B - x_A| = |4 - (-6)| = |4 + 6| = 10 \]

3. **Determine the Midpoint of AB**:
- The midpoint \(P\) of segment AB can be calculated as:
\[ P = \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) = \left(\frac{-6 + 4}{2}, \frac{7 + 7}{2}\right) = \left(\frac{-2}{2}, 7\right) = (-1, 7) \]

4. **Find the Center of the Circle**:
- The center \(O\) of the circle lies on the line defined by the equation \(3y = x + 10\).
- Substitute \(x = -1\) into the equation to find \(y\):
\[ 3y = -1 + 10 \implies 3y = 9 \implies y = 3 \]
- Therefore, the center \(O\) is at \((-1, 3)\).

5. **Determine the Coordinates of Point C**:
- Since \(O\) is the midpoint of diagonal AC, we can use the midpoint formula:
\[ O = \left(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}\right) \]
- Setting up the equations:
\[ -1 = \frac{-6 + x_C}{2} \quad \text{and} \quad 3 = \frac{7 + y_C}{2} \]
- Solving for \(x_C\):
\[ -2 = -6 + x_C \implies x_C = 4 \]
- Solving for \(y_C\):
\[ 6 = 7 + y_C \implies y_C = -1 \]
- Thus, the coordinates of point C are \(C = (4, -1)\).

6. **Determine the Coordinates of Point D**:
- Since \(D\) is directly opposite to \(B\) in the rectangle, and \(B\) has coordinates \((4, 7)\), the coordinates of \(D\) will be:
\[ D = (-6, -1) \]

7. **Calculate the Length of BC**:
- The length of segment BC can be calculated as:
\[ \text{Length of BC} = |y_B - y_C| = |7 - (-1)| = |7 + 1| = 8 \]

8. **Calculate the Area of Rectangle ABCD**:
- The area \(A\) of rectangle ABCD is given by:
\[ A = \text{Length} \times \text{Breadth} = 10 \times 8 = 80 \text{ square units} \]

### Final Answer:
The area of rectangle ABCD is \(80\) square units.