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:
Show Hint
- \( \sqrt{14} \)
- \( 5\sqrt{7} \)
- \( 3\sqrt{7} \)
- \( 2\sqrt{14} \)
The Correct Option is A
Solution and Explanation
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).
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