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
a) Kituo cha duara. Equation ya duara iliyotolewa ni x^2 + y^2 - 3x + 4y = 0. Equation ya jumla ya duara ni x^2 + y^2 + 2gx + 2fy + c = 0, ambapo kituo cha duara ni (-g, -f). Step 1: Linganisha coefficients za x na y kutoka equation uliyopewa na equation ya jumla. 2gx = -3x 2g = -3 g = -(3)/(2) 2fy = 4y 2f = 4 f = 2 Step 2: Tumia maadili ya g na f kupata kituo cha duara. Kituo ni (-g, -f). (-g, -f) = (-(-(3)/(2)), -2) = ((3)/(2), -2) Au kwa desimali: (1.5, -2). Kituo cha duara ni ((3)/(2), -2) au (1.5, -2). b) Eneo la duara. Kuna uwezekano kwamba "area of triangle" ni makosa ya kuandika na inapaswa kuwa "area of circle", hasa kwa sababu swali linaomba jibu "in terms of ". Tutachukulia kuwa swali linauliza eneo la duara. Equation iliyotolewa katika sehemu b) ni x^2 - 2ax + a^2 + y^2 - 2by + b^2 = r^2, ambayo inaweza kuandikwa kama (x-a)^2 + (y-b)^2 = r^2. Hii ni equation ya duara yenye kituo (a,b) na radius r. Tutatumia duara ileile kutoka sehemu a). Step 3: Andika upya equation ya duara kutoka sehemu a) katika fomu ya kawaida (x-a)^2 + (y-b)^2 = r^2 kwa kukamilisha mraba. Equation: x^2 + y^2 - 3x + 4y = 0 Panga upya maneno ya x na y: (x^2 - 3x) + (y^2 + 4y) = 0 Kamilisha mraba kwa maneno ya x na y: (x^2 - 3x + (-(3)/(2))^2) + (y^2 + 4y + ((4)/(2))^2) = (-(3)/(2))^2 + ((4)/(2))^2 (x - (3)/(2))^2 + (y + 2)^2 = (9)/(4) + 4 (x - (3)/(2))^2 + (y + 2)^2 = (9)/(4) + (16)/(4) (x - (3)/(2))^2 + (y + 2)^2 = (25)/(4) Step 4: Tambua r^2 kutoka equation iliyokamilishwa. Kutoka equation (x - 1.5)^2 + (y + 2)^2 = (25)/(4), tunaona kwamba r^2 = (25)/(4). Step 5: Kokotoa eneo la duara. Eneo la duara ni A = r^2. A = ((25)/(4)) A = (25)/(4) Eneo la duara ni (25)/(4). Tuma swali linalofuata 📸