Final answer:
The centre of the circle is (1, -1), the radius is √(52)/2, and the equation of the circle is (x - 1)^2 + (y + 1)^2 = 13.
Step-by-step explanation:
The student has asked to find the centre, radius and the equation of the circle that has the line segment joining the points (-1, 2) and (3, -4) as its diameter. The centre of the circle is the midpoint of the given diameter. To find the midpoint, we use the midpoint formula which is ((x1 + x2)/2, (y1 + y2)/2). Plugging in our points, we have ((-1 + 3)/2, (2 - 4)/2) which simplifies to (1, -1). This is the centre of our circle.
The radius of the circle is half the distance of the diameter. To find the distance between the two points, we use the distance formula which is √((x2 - x1)^2 + (y2 - y1)^2). For our points, the calculation is √((3 - (-1))^2 + (-4 - 2)^2) which simplifies to √(16 + 36) or √(52). The radius is therefore half of this value, √(52)/2.
The equation of the circle with center (h, k) and radius r is given by (x - h)^2 + (y - k)^2 = r^2. Substituting our centre (1, -1) and radius √(52)/2 into this formula, we get (x - 1)^2 + (y + 1)^2 = (√(52)/2)^2, which simplifies to (x - 1)^2 + (y + 1)^2 = 13.