The equation of a circle which passes through the points (2,3) and (4,5) and whose center lies on the straight line y - 4x + 3 = 0 is

Step-by-step solution

Let the center of the circle be (h, k) . Since the center lies on the line y - 4x + 3 = 0 , we have: k - 4h + 3 = 0 k = 4h - 3 The circle passes through (2,3) and (4,5) , so the distance from the center to both points is equal to the radius r . (h - 2)^2 + (k - 3)^2 = (h - 4)^2 + (k - 5)^2 Expanding both sides: h^2 - 4h + 4 + k^2 - 6k + 9 = h^2 - 8h + 16 + k^2 - 10k + 25 Simplifying the equation: 4h + 4k = 28 h + k = 7 Substituting k = 4h - 3 into the above equation: h + 4h - 3 = 7 5h = 10 h = 2 Then, k = 4(2) - 3