(1)<\/strong>\u3000\\(u\\) \u3092\u5b9f\u6570\u3068\u3057,\r\n\\(C\\) \u4e0a\u306e\u70b9\\(\\left( u , b \\sqrt{1 +\\dfrac{u^2}{a^2}} \\right)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092, \\(a , b , u\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(C\\) \u4e0a\u306e\u7570\u306a\u308b \\(2\\) \u70b9\u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u4ea4\u70b9\u306e\u5168\u4f53\u304b\u3089\u306a\u308b\u9818\u57df\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000(2)<\/strong> \u306e\u9818\u57df\u306b\u3042\u308b\u70b9 \\(( p , q )\\) \u306b\u3064\u3044\u3066,\r\n\u70b9 \\(( p , q )\\) \u3092\u901a\u308b \\(C\\) \u306e\u63a5\u7dda\u306e\u63a5\u70b9\u3092\u3059\u3079\u3066\u901a\u308b\u76f4\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092, \\(a , b , p , q\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u66f2\u7dda \\(C\\) \u306f\u53cc\u66f2\u7dda\u3067, \\(\\dfrac{x^2}{a^2} -\\dfrac{y^2}{b^2} = -1\\) . (2)<\/strong><\/p>\r\n (1)<\/strong> \u3067\u6c42\u3081\u305f\u63a5\u7dda\u304c, \u70b9 \\(( X , Y )\\) \u3092\u901a\u308b\u3068\u304d\r\n\\[\r\nY = \\dfrac{b ( uX +a^2 )}{a \\sqrt{a^2 +u^2}} \\quad ... [1]\r\n\\]\r\n\u3053\u308c\u3092\u307f\u305f\u3059 \\(u\\) \u304c \\(2\\) \u3064\u3042\u308b\u305f\u3081\u306e\u6761\u4ef6\u3092\u8003\u3048\u308c\u3070\u3088\u3044. (3)<\/strong><\/p>\r\n (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \u70b9 \\(( p , q )\\) \u3092\u901a\u308b\u63a5\u7dda\u306f \\(2\\) \u672c\u3042\u308a, \u305d\u308c\u305e\u308c\u306e\u63a5\u70b9\u3092 \\(( x_1 , y_1 ) , ( x_2 , y_2 )\\) \u3068\u304a\u304f\u3068, \u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\r\n\\dfrac{x_1}{a^2} x -\\dfrac{y_1}{b^2} y = -1 \\ , \\ \\dfrac{x_2}{a^2} x -\\dfrac{y_2}{b^2} y = -1\r\n\\]\r\n\u3053\u308c\u304c, \u70b9 \\(( p , q )\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\r\n\\left\\{ \\begin{array}{l} \\dfrac{p x_1}{a^2} -\\dfrac{q y_1}{b^2} = -1 \\\\ \\dfrac{p x_2}{a^2} -\\dfrac{q y_2}{b^2} = -1 \\end{array} \\right.\r\n\\]\r\n\u3053\u308c\u306f, \u76f4\u7dda \\(\\dfrac{p}{a^2} x -\\dfrac{q}{b^2} y = -1\\) \u304c \\(2\\) \u70b9 \\(( x_1 , y_1 ) , ( x_2 , y_2 )\\) \u3092\u901a\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(C\\) \u306e\u5f0f\u3092 \\(y = f(x)\\) \u3068\u304a\u304f.\r\n\\[\r\nf'(x) = \\dfrac{2x}{a^2} \\cdot \\dfrac{1}{2 \\sqrt{1 +\\dfrac{x^2}{a^2}}} = \\dfrac{xb}{a \\sqrt{a^2 +x^2}}\r\n\\]\r\n\u306a\u306e\u3067, \u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = \\dfrac{ub}{a \\sqrt{a^2 +u^2}} ( x-u ) +b \\sqrt{1 +\\dfrac{u^2}{a^2}} \\\\\r\n& = \\dfrac{ub x -u^2 b +b ( a^2 +u^2 )}{a \\sqrt{a^2 +x^2}} \\\\\r\n& = \\dfrac{b ( ux +a^2 )}{a \\sqrt{a^2 +x^2}}\r\n\\end{align}\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\underline{y = \\dfrac{b ( ux +a^2 )}{a \\sqrt{a^2 +u^2}}}\r\n\\]\r\n
\r\n[1] \u306e\u53f3\u8fba\u3092 \\(g(u)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\ng'(u) & = \\dfrac{b}{a} \\cdot \\dfrac{X \\sqrt{u^2 +a^2} -\\dfrac{2u ( Xu +a^2 )}{2 \\sqrt{u^2 +a^2}}}{u^2 +a^2} \\\\\r\n& = \\dfrac{b}{a} \\cdot \\dfrac{X ( u^2 +a^2 ) -u ( Xu +a^2 )}{( u^2 +a^2 )^{\\frac{3}{2}}} \\\\\r\n& = \\dfrac{ab ( X-u )}{( u^2 +a^2 )^{\\frac{3}{2}}}\r\n\\end{align}\\]\r\n\\(g'(u) = 0\\) \u3092\u3068\u304f\u3068, \\(u = X\\) .
\r\n\\[\r\ng(X) = b \\sqrt{1 +\\dfrac{X^2}{a^2}} = f(X)\r\n\\]\r\n\u307e\u305f\r\n\\[\r\ng(u) = \\dfrac{b}{a} \\cdot \\dfrac{\\dfrac{u}{|u|} X +\\dfrac{a^2}{|u|}}{\\sqrt{1 +\\dfrac{a^2}{u^2}}}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\lim_{u \\rightarrow \\infty} g(u) = \\dfrac{b}{a} X \\ , \\ \\displaystyle\\lim_{u \\rightarrow -\\infty} g(u) = -\\dfrac{b}{a} X\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(g(u)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} u & ( -\\infty ) & \\cdots & X & \\cdots & ( \\infty ) \\\\ \\hline g'(u) & & + & 0 & - & \\\\ \\hline g(u) & \\left( -\\dfrac{b}{a} X \\right) & \\nearrow & f(X) & \\searrow & \\left( \\dfrac{b}{a} X \\right) \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\nY \\gt -\\dfrac{b}{a} X \\ \\text{\u304b\u3064} \\ Y \\gt \\dfrac{b}{a} X \\ \\text{\u304b\u3064} \\ Y \\lt f(X)\r\n\\]\r\n\u3053\u308c\u3092, \u56f3\u793a\u3059\u308b\u3068\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n![\"iks20210301\"](\"\/nyushi\/wp-content\/uploads\/iks20210301.svg\")
\r\n
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5f0f\u306f\r\n\\[\r\n\\underline{\\dfrac{p}{a^2} x -\\dfrac{q}{b^2} y = -1}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a , b\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3057, \u66f2\u7dda \\(C : y = b \\sqrt{1 +\\dfrac{x^2}{a^2}}\\) \u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(u\\) \u3092\u5b9f\u6570\u3068\u3057, \\(C\\) \u4e0a\u306e … \u7d9a\u304d\u3092\u8aad\u3080