Let the line x+y=1 meet the circle x2+y2=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 ADBC is equal to:
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Step-by-Step Solution
Step 1: Find the midpoint of chord AB
The given circle is x2+y2=4, which means its center is at the origin (0,0) and its radius is r=2. The line x+y=1 is a chord of this circle. The line perpendicular to the chord AB and passing through its midpoint M must pass through the center of the circle. This is a fundamental property of circles: the perpendicular bisector of any chord passes through the center. Since the line x+y=1 has a slope of −1, the perpendicular line passing through the center (0,0) will have a slope of 1. The equation of this perpendicular line is y=x. The midpoint M of the chord AB is the intersection of the line x+y=1 and the line y=x. Substituting y=x into x+y=1 gives x+x=1, so 2x=1, which means x=1/2. Therefore, y=1/2, and the midpoint M is (1/2,1/2).
Step 2: Calculate the length of chord AB
The distance from the center (0,0) to the midpoint M(1/2,1/2) is OM=(1/2)2+(1/2)2=1/4+1/4=1/2=1/2. We can use the Pythagorean theorem in triangle OMA, where OA is the radius r=2. So, AM2=OA2−OM2=22−(1/2)2=4−1/2=7/2. Thus, AM=7/2. The length of the chord AB is 2×AM=2×7/2=4×7/2=14.
Step 3: Identify the quadrilateral ADBC
The line perpendicular to AB passing through its midpoint M also passes through the center of the circle. This line intersects the circle at points C and D. Since AB is a chord and CD is a diameter (because it passes through the center and is perpendicular to AB), the quadrilateral ADBC has diagonals AB and CD that are perpendicular bisectors of each other. This means ADBC is a rhombus. Furthermore, since CD is a diameter, C and D are diametrically opposite points. The line CD is the line y=x. The points C and D are the intersections of y=x and x2+y2=4. Substituting y=x gives x2+x2=4, so 2x2=4, x2=2, x=±2. So C=(2,2) and D=(−2,−2). The length of the diagonal CD is 2r=4.
Step 4: Calculate the area of the rhombus
The area of a rhombus is given by half the product of its diagonals. Here, the diagonals are AB and CD. We found AB=14 and CD=4.
Step 5: Substitute values and find the area
Substituting the lengths of the diagonals AB=14 and CD=4 into the area formula, we get Area=21×14×4=214.