Let C be a circle having centre in the first quadrant and touching the x -axis at a distance of 3 units from the origin. If the circle C has an intercept of length 6√3 on y -axis, then the length of the chord of the circle C on the line x - y = 3 is :

Step-by-step solution

Let the centre of the circle be (3, r) since it touches the x -axis at (3, 0) and lies in the first quadrant. The radius of the circle is r . The length of the intercept on the y -axis is given by 2√(r^2 - d^2) , where d is the distance of the centre from the y -axis. Here, d = 3 . Given that the intercept on the y -axis is 6√3 : 2√(r^2 - 3^2) = 6√3 √(r^2 - 9) = 3√3 r^2 - 9 = 27 r^2 = 36 ⇒ r = 6 The centre of the circle is (3, 6) and its radius is 6 . The given line is x - y - 3 = 0 . The perpendicular distance p from the centre (3, 6) to the line is: p = |3 - 6 - 3| √(1^2 + (-1)^2) = 6 √2 = 3√2 The length of the chord is 2√(r^2 - p^2) . Length of the chord = 2 36 - (3√2)^2 = 2√(36 - 18) = 2√(18) = 6√2 Answer: 6√2