(a) Find the equation of a circle with a radius of 5 units and its centre at the point (−1, 2).(b) What is the length of the intercept cut off by this circle on the x-axis?(c) Find the length of the tangent to this circle from the point (4, 6).

(a) The general equation of a circle is given by (x-h)² + (y-k)² = r² where (h,k) is the center of the circle and r is the radius. Given that the center is (-1,2) and the radius is 5, we substitute these values into the equation to get:

(x - (-1))² + (y - 2)² = 5² => (x + 1)² + (y - 2)² = 25

(b) The x-intercepts of the circle are the points where the circle intersects the x-axis. At these points, y = 0. Substituting y = 0 into the equation of the circle gives:

(x + 1)² + (0 - 2)² = 25 => (x + 1)² + 4 = 25 => (x + 1)² = 21 => x + 1 = ± √21 => x = -1 ± √21

The length of the intercept cut off by the circle on the x-axis is the distance between these two points, which is 2√21 units.

(c) The length of the tangent from a point (x1, y1) to a circle with center (h, k) and radius r is given by √((x1 - h)² + (y1 - k)² - r²). Substituting the given values gives:

Length = √((4 - (-1))² + (6 - 2)² - 5²) = √((5)² + (4)² - 5²) = √(16) = 4 units.