Equation of a Circle with Diameter Endpoints Calculator

To find a circle's equation from two diameter endpoints, (x1, y1) and (x2, y2), the calculator applies fundamental analytic geometry principles. First, the midpoint formula determines the center of the circle (h, k): h = (x1 + x2) / 2 and k = (y1 + y2) / 2. Next, the distance formula calculates the total diameter length, which is then halved to find the radius (r). Finally, these values are substituted into the standard circle equation form: (x - h)^2 + (y - k)^2 = r^2, and expanded into the general form Ax^2 + By^2 + Dx + Ey + F = 0.

Let us find the equation of a circle whose diameter endpoints are given by (2, 3) and (6, 7). First, compute the center coordinates by averaging the x and y values: h = (2 + 6) / 2 = 4, and k = (3 + 7) / 2 = 5. The center of our circle is at (4, 5). Second, calculate the radius by finding the distance between the two points and dividing by 2: r = sqrt((6 - 2)^2 + (7 - 3)^2) / 2 = sqrt(16 + 16) / 2 = sqrt(32) / 2 = 5.656 / 2 = 2.828, meaning r^2 = 8. Thus, the standard equation is (x - 4)^2 + (y - 5)^2 = 8. Expanding this yields the general form: x^2 + y^2 - 8x - 10y + 33 = 0.

Always double-check the order of your coordinate points; while subtraction order is squared away during distance calculations, mixing up x and y values will misplace your center point. Remember that the radius is always half of the total diameter length, not the full distance between the two given endpoints. When converting from standard form to general form, expand binomials carefully to avoid sign errors in your linear coefficients.