This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Right Hurry, let's go. a) Find the equation of the circle. Step 1: Find the center of the circle. The center (h, k) is the midpoint of the diameter's endpoints (-3, 5) and (7, 9). h = (-3+7)/(2) = (4)/(2) = 2 k = (5+9)/(2) = (14)/(2) = 7 The center of the circle is (2, 7). Step 2: Find the radius of the circle. The radius r is the distance from the center (2, 7) to one of the endpoints, for example, (7, 9). r = sqrt((7-2)^2 + (9-7)^2) r = sqrt((5)^2 + (2)^2) r = sqrt(25 + 4) r = sqrt(29) The square of the radius is r^2 = 29. Step 3: Write the equation of the circle. The standard equation of a circle is (x-h)^2 + (y-k)^2 = r^2. Substituting the center (2, 7) and r^2 = 29: (x-2)^2 + (y-7)^2 = 29 b) Find the equation of the normal to the circle at (3, 4). A normal to a circle always passes through its center. Therefore, the normal line passes through the center (2, 7) and the given point (3, 4). (Note: The point (3, 4) is not on the circle, as (3-2)^2 + (4-7)^2 = 1^2 + (-3)^2 = 1+9=10 ≠ 29. We will find the equation of the line passing through the center (2, 7) and the point (3, 4).) Step 1: Find the slope of the normal line. Using the points (2, 7) and (3, 4): m = (4-7)/(3-2) = (-3)/(1) = -3 Step 2: Write the equation of the normal line. Using the point-slope form y - y_1 = m(x - x_1) with point (3, 4) and slope m = -3: y - 4 = -3(x - 3) y - 4 = -3x + 9 y = -3x + 9 + 4 y = -3x + 13 y = -3x + 13 Got more? Send 'em 📸