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 mid point 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 areas of quadrilaterals formed by intersection points on a circle, use coordinate geometry to find the points and apply the area formula for polygons.

Answer 1

The Correct Option is A

Step 1: The given circle equation is \( x^2 + y^2 = 4 \). The equation of the line \( x + y = 1 \) will intersect the circle at two points, which we need to find. 
Step 2: Solve the system of equations \( x + y = 1 \) and \( x^2 + y^2 = 4 \) to find the points of intersection A and B. 
Step 3: The line perpendicular to \( AB \) passing through the midpoint of \( AB \) will intersect the circle again at points C and D. Use geometric properties of the circle and the perpendicular bisector to find the area of quadrilateral ABCD. 
Step 4: After solving for the coordinates of points A, B, C, and D, calculate the area of quadrilateral ABCD, which evaluates to \( \sqrt{14} \). Thus, the correct answer is (1).