Let C be a circle given by the equation x^2 + y^2 = 25 . A chord AB of the circle lies on the line 3x - 4y + 15 = 0 . A second chord CD is parallel to AB , has a length of 6 units, and lies on the opposite side of the centre from AB . If the endpoints of the chords are joined in order to form a quadrilateral ACDB , then the area of this quadrilateral (in square units) is

Step-by-step solution

The radius of the circle x^2 + y^2 = 25 is r = 5 . The perpendicular distance of the chord AB from the centre (0, 0) is: d₁ = |3(0) - 4(0) + 15| 3^2 + (-4)^2 = 15 5 = 3 The length of the chord AB is: AB = 2 r^2 - d₁^2 = 2 25 - 9 = 2(4) = 8 The length of the chord CD is given as 6 . Its perpendicular distance from the centre is: d₂ = r^2 - ( CD 2 )^2 = 25 - 3^2 = 4 Since CD is parallel to AB and lies on the opposite side of the centre, the perpendicular distance between the two chords (which is the height of the tra