A rectangle is inscribed in a circle with a diameter lying along the line . If the two adjacent vertices of the rectangle are and , then the area of the rectangle (in sq units) is
- A.72
- B.84
- C.98
- D.56
Show answer and solution
Correct option: B
Solution
The two adjacent vertices (-8, 5) and (6, 5) give one side of length 14. Since the diameter lies along 3y = x + 7, the center of the circle lies on this line. The center is also the midpoint of the diagonal of the rectangle, and by symmetry it lies on the perpendicular bisector of the side from (-8,5) to (6,5), which is x = -1. Substituting x = -1 into 3y = x + 7: 3y = 6, so y = 2. Center = (-1, 2).
The other two vertices are reflections of (-8,5) and (6,5) across the center (-1,2): (-1-(-8-(-1)), 2-(5-2)) = (6, -1) and (-1-(6-(-1)), -1) = (-8, -1). So the rectangle has vertices (-8,5), (6,5), (6,-1), (-8,-1).
Width = 14, Height = 5-(-1) = 6. Area = 14 × 6 = 84.
The other two vertices are reflections of (-8,5) and (6,5) across the center (-1,2): (-1-(-8-(-1)), 2-(5-2)) = (6, -1) and (-1-(6-(-1)), -1) = (-8, -1). So the rectangle has vertices (-8,5), (6,5), (6,-1), (-8,-1).
Width = 14, Height = 5-(-1) = 6. Area = 14 × 6 = 84.
[JEE Main 2019 · 9 April · Shift I]