Question

Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the midpoint of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to:

• A. \( \sqrt{14} \)
• B. \( 5\sqrt{7} \)
• C. \( 3\sqrt{7} \)
• D. \( 2\sqrt{14} \)

Hint:

To find the area of a quadrilateral formed by intersecting curves, use geometric properties and formulas for the area of triangles and rectangles.

Answer 1

The Correct Option is A

- First, find the points of intersection of the line and the circle.
- Then, find the midpoint of the chord and the points where the perpendicular line intersects the circle.
- Using geometry, calculate the area of the quadrilateral formed by points A, B, C, and D. The area is \( \sqrt{14} \).