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 :
C. 6 2
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
Related: Mathematics — Circle · All PYQ Banks