数学 – 図形と数式の関係 - 平面図形と式 – 円と軌跡(三角形の外接円, 外心)

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数学読本〈2〉簡単な関数/平面図形と式/指数関数・対数関数/三角関数 (松坂和夫(著)、岩波書店)の第6章(図形と数式の関係 - 平面図形と式)、6.3(円と軌跡)、三角形の外接円、問21.を解いてみる。

問21.

x^{2} + y^{2} + l x + m y + n = 0

$\begin{array}{l} x^{2} + y^{2} + l x + m y + n = 0 \\ 16 + 36 + 4 l + 6 m + n = 0 \\ n = - 4 l - 6 m - 52 \\ 9 + 25 - 3 l + 5 m - 4 l - 6 m - 52 = 0 \\ - 18 - 7 l - m = 0 \\ m = - 7 l - 18 \\ n = - 4 l + 42 l + 108 - 52 = 38 l + 56 \\ 1 + 9 + l - 3 m + n = 0 \\ 1 + 9 + l + 21 l + 54 + 38 l + 56 = 0 \\ 60 l = - 120 \\ l = - 2 \\ m = - 4 \\ n = - 20 \\ x^{2} + y^{2} - 2 x - 4 y - 20 = 0 \\ {(x - 1)}^{2} + {(y - 2)}^{2} = 5^{2} \\ (1, 2) \end{array}$
$\begin{array}{l} 49 - 7 m + n = 0 \\ n = 7 m - 49 \\ 121 + 16 + 11 l - 4 m + n = 0 \\ 121 + 16 + 11 l - 4 m + 7 m - 49 = 0 \\ m = \frac{- 88 - 11 l}{3} \\ n = 7 m - 49 = \frac{- 7 (88 + 11 l)}{3} - 49 \\ 25 + 64 + 5 l + 8 m + n = 0 \\ 25 + 64 + 5 l + \frac{- 8 (88 + 11 l)}{3} + \frac{- 7 (88 + 11 l)}{3} - 49 = 0 \\ 40 + 5 l - \frac{15 (88 + 11 l)}{3} = 0 \\ 40 + 5 l - 440 - 55 l = 0 \\ 50 l = - 400 \\ l = - 8 \\ m = 0 \\ n = - 49 \\ x^{2} + y^{2} - 8 x - 49 = 0 \\ {(x - 4)}^{2} + y^{2} = 65 \\ (4, 0) \end{array}$

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