Question

Find the equation of the circle passing through three points(4, 7), (5, 6), and (1, 8). Also, find the coordinates of the point of intersection of the tangents to the circle at the points where it is cut by the straight line 5x+ y + 17 = 0.

Answer 1

To find the equation of the circle passing through three points, use the general form of the equation: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is its radius. Set up a system of equations using the given points, solve the system to find the values of h, k, and r, and substitute the values into the general form of the equation.

Step-by-step explanation:

To find the equation of a circle passing through three points, we need to use the general form of the equation: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is its radius.

Step 1: Use the given points to set up a system of equations:

(4 - h)^2 + (7 - k)^2 = r^2

(5 - h)^2 + (6 - k)^2 = r^2

(1 - h)^2 + (8 - k)^2 = r^2

Step 2: Solve the system of equations to find the values of h, k, and r.

Step 3: Substitute the values of h, k, and r into the general form of the equation to get the final equation of the circle.