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.
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Huu hapa ni utatuzi wa swali la 11. Swali la 11: Takwimu iliyo hapa chini ni sehemu ya duara yenye pointi A, B na C kwenye sehemu hiyo. Bainisha kuratibu za kituo cha duara (alama 4). Pointi zilizopewa ni: A(-3, 5) B(0, 2) C(-3, -1) Step 1: Fafanua kuratibu za kituo cha duara. Acha kituo cha duara kiwe (h, k). Umbali kutoka kituo hadi pointi yoyote kwenye duara ni sawa (radius). Hivyo, CA^2 = CB^2 = CC^2. Step 2: Unda milinganyo kwa kutumia fomula ya umbali. Tunatumia fomula ya umbali d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2. Kwanza, linganisha umbali wa mraba kutoka kituo hadi A na B (CA^2 = CB^2): (h - (-3))^2 + (k - 5)^2 = (h - 0)^2 + (k - 2)^2 (h + 3)^2 + (k - 5)^2 = h^2 + (k - 2)^2 Panua milinganyo: h^2 + 6h + 9 + k^2 - 10k + 25 = h^2 + k^2 - 4k + 4 Ondoa h^2 na k^2 kutoka pande zote mbili: 6h - 10k + 34 = -4k + 4 Panga upya ili kupata mlinganyo wa kwanza: 6h - 10k + 4k = 4 - 34 6h - 6k = -30 Gawanya kwa 6: h - k = -5 (Mlinganyo 1) Pili, linganisha umbali wa mraba kutoka kituo hadi B na C (CB^2 = CC^2): (h - 0)^2 + (k - 2)^2 = (h - (-3))^2 + (k - (-1))^2 h^2 + (k - 2)^2 = (h + 3)^2 + (k + 1)^2 Panua milinganyo: h^2 + k^2 - 4k + 4 = h^2 + 6h + 9 + k^2 + 2k + 1 Ondoa h^2 na k^2 kutoka pande zote mbili: -4k + 4 = 6h + 2k + 10 Panga upya ili kupata mlinganyo wa pili: 0 = 6h + 2k + 4k + 10 - 4 0 = 6h + 6k + 6 Gawanya kwa 6: h + k = -1 (Mlinganyo 2) Step 3: Tatua mfumo wa milinganyo. Tuna milinganyo miwili ya mstari: 1) h - k = -5 2) h + k = -1 Ongeza Mlinganyo 1 na Mlinganyo 2: (h - k) + (h + k) = -5 + (-1) 2h = -6 h = -3 Step 4: Tafuta thamani ya k. Badilisha h = -3 kwenye Mlinganyo 2: -3 + k = -1 k = -1 + 3 k = 2 Kuratibu za kituo cha duara ni (-3, 2). (-3, 2) That's 2 down. 3 left today — send the next one.