Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line x + y = 1 is 14 , then the square of the radius of C is _______.
A. A
Let the equation of the circle be x^2 + y^2 + 2gx + 2fy + c = 0. Since the circle intersects the coordinate axes at exactly three points, it must pass through the origin and intersect each axis at one other distinct point. Thus, c = 0. The lengths of the intercepts on the x-axis and y-axis are 2 g^2 - c = 2|g| and 2 f^2 - c = 2|f|. Given that the intercepts are equal, we have |g| = |f|. The centre of the circle is (-g, -f). Since it lies in the first quadrant, -g > 0 and -f > 0. Therefore, g = f = -a for some a > 0. The centre is (a, a) and the radius is r = a^2 + a^2 = a 2 . The distance d from the centre (a, a) to the line x + y - 1 = 0 is given by: d = |a + a - 1| 1^2 + 1^2 = |2a - 1| 2 The length of the chord on the given line is 2 r^2 - d^2 = 14 . Squaring both sides, we get: 4(r^2 - d^2) = 14 r^2 - d^2 = 7 2 Substituting r^2 = 2a^2 and d^2 = (2a - 1)^2 2 : 2a^2 - (2a - 1)^2 2 = 7 2 4a^2 - (4a^2 - 4a + 1) = 7 4a - 1 = 7 4a = 8 a = 2 The square of the radius of the circle is r^2 = 2a^2 = 2(2)^2 = 8. Answer: 8
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