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 ADBC is equal to :
Step-by-Step Solution
Key Concept: Apply the core result for circle equations and tangents and simplify using the given constraints.
(3) By solving x = y with circle We get C(\sqrt2, \sqrt2) D(-\sqrt2, -\sqrt2) By solving x + y = 1 with circle x + y 2 2 = 4 we set 1 + \sqrt7 1 - \sqrt7 A( , ) 2 2 1 - \sqrt7 1 + \sqrt7 & B( , ) 2 2 \therefore Area of Quadrilateral ACBD = 2 \times Area of △BCD ∣ \sqrt2 \sqrt2 1∣ ∣ ∣ 1 1-\sqrt7 1+\sqrt7 = 2 \times ∣ 1∣ 2 2 2 ∣ ∣ ∣ -\sqrt2 -\sqrt2 1∣ = 2\sqrt14
Correct Answer: 3