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
Step 1: Find the center of the circle. The equation of the circle is 3x^2 + 3y^2 + 5x - 3y - 22 = 0. Divide by 3 to get the standard form x^2 + y^2 + (5)/(3)x - y - (22)/(3) = 0. Comparing this to the general equation x^2 + y^2 + 2gx + 2fy + c = 0, we have: 2g = (5)/(3) g = (5)/(6) 2f = -1 f = -(1)/(2) The center of the circle is (-g, -f) = (-(5)/(6), (1)/(2)). Step 2: Find the equation of the normal (Part a). The normal to a circle at any point passes through the center of the circle. We need to find the equation of the line passing through the given point (2,1) and the center (-(5)/(6), (1)/(2)). First, calculate the slope m of the normal: m = (y_2 - y_1)/(x_2 - x_1) = (1 - 1)/(2)2 - (-(5)/(6)) = (1)/(2)2 + (5)/(6) = (1)/(2)(12)/(6) + (5)/(6) = (1)/(2)(17)/(6) = (1)/(2) × (6)/(17) = (3)/(17) Now, use the point-slope form y - y_1 = m(x - x_1) with the point (2,1): y - 1 = (3)/(17)(x - 2) 17(y - 1) = 3(x - 2) 17y - 17 = 3x - 6 3x - 17y + 11 = 0 The equation of the normal is 3x - 17y + 11 = 0. Step 3: Calculate the length of the diameter (Part b). The normal cuts the circle at two points, (2,1) and (-(11)/(3), 0). Since the normal passes through the center, these two points are the endpoints of a diameter. Use the distance formula to find the length of the diameter D: D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) D = sqrt((-(11)/(3) - 2)^2 + (0 - 1)^2) D = sqrt((-(11)/(3) - (6)/(3))^2 + (-1)^2) D = sqrt((-(17)/(3))^2 + 1^2) D = sqrt((289)/(9) + 1) D = sqrt((289)/(9) + (9)/(9)) D = sqrt((298)/(9)) D = sqrt(298)3 The length of the diameter is sqrt(298)3. Send me the next one 📸