(1)<\/strong>\u3000\\(v^2\\) \u3092 \\(a , b , u\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(V\\) \u306e\u6700\u5927\u5024\u3092, \\(a , b\\) \u3092\u7528\u3044\u3066\u8868\u305b. \u307e\u305f, \u305d\u306e\u3068\u304d\u306e \\(C\\) \u306e\u65b9\u7a0b\u5f0f\u3092 \\(a , b\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C\\) \u4e0a\u306e\u70b9 P \\((p,q)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\r\n\\dfrac{px}{v^2} +\\dfrac{qy}{u^2} = 1\r\n\\]\r\n\u3053\u308c\u3068 \\(\\ell\\) \u306e\u5f0f\u3068\u6bd4\u8f03\u3057\u3066\r\n\\[\\begin{align}\r\n\\dfrac{p}{v^2} = \\dfrac{1}{a} & , \\ \\dfrac{q}{u^2} = \\dfrac{1}{b} \\\\\r\n\\text{\u2234} \\quad p = \\dfrac{v^2}{a} & , \\ q = \\dfrac{u^2}{b} \\quad ... [1]\r\n\\end{align}\\]\r\n\u307e\u305f, \u70b9 P \u306f \\(C\\) \u4e0a\u306b\u3042\u308b\u306e\u3067, [1] \u3092\u7528\u3044\u3066\r\n\\[\\begin{gather}\r\n\\dfrac{\\left( \\frac{v^2}{a} \\right)^2}{u^2} +\\dfrac{\\left( \\frac{u^2}{b} \\right)}{v^2} = 1 \\\\\r\n\\dfrac{v^2}{a^2} +\\dfrac{u^2}{b^2} = 1 \\\\\r\n\\text{\u2234} \\quad v^2 = \\underline{b^2 \\left( 1 -\\dfrac{u^2}{a^2} \\right)}\r\n\\end{gather}\\]\r\n (2)<\/strong><\/p>\r\n \\(u\\) \u306e\u53d6\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f, \\(0 \\lt u \\lt a\\) .
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
![\"yokokoku_r_2007_02_01\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/yokokoku_r_2007_02_01.png\")
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\r\n\u56de\u8ee2\u4f53\u306f, \u534a\u5f84 \\(v\\) \u306e\u7403\u3092 \\(x\\) \u8ef8\u65b9\u5411\u306b \\(\\dfrac{u}{v}\\) \u500d\u3057\u305f\u7acb\u4f53\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nV & = \\dfrac{4 \\pi v^3}{3} \\cdot \\dfrac{u}{v} \\\\\r\n& = \\dfrac{4 \\pi}{3} u \\cdot b^2 \\left( 1 -\\dfrac{u^2}{a^2} \\right) \\\\\r\n& = \\dfrac{4 \\pi b^2}{3 a^2} \\underline{u ( a^2-u^2 )} _ {[ \\text{A} ]}\r\n\\end{align}\\]\r\n[A] \u3092 \\(f(u)\\) \u3068\u304a\u3044\u3066, \\(f(u)\\) \u304c\u6700\u5927\u306e\u3068\u304d\u3092\u8003\u3048\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nf'(u) & = a^2 -3u^2 \\\\\r\n& = \\left( a +\\sqrt{3} u \\right) \\left( a -\\sqrt{3} u \\right)\r\n\\end{align}\\]\r\n\\(f'(u) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nu = \\dfrac{a}{\\sqrt{3}}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(u)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a\u3067\r\n\\[\r\n\\begin{array}{c|ccccc} u & (0) & \\cdots & \\frac{a}{\\sqrt{3}} & \\cdots & (a) \\\\ \\hline f'(u) & & + & 0 & - & \\\\ \\hline f(u) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u6700\u5927\u5024\u306f\r\n\\[\\begin{align}\r\nf \\left( \\dfrac{a}{\\sqrt{3}} \\right) & = \\dfrac{a}{\\sqrt{3}} \\cdot \\dfrac{2 a^2}{3} \\\\\r\n& = \\dfrac{2 \\sqrt{3} a^3}{9}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{4 \\pi b^2}{3 a^2} f \\left( \\dfrac{a}{\\sqrt{3}} \\right) = \\underline{\\dfrac{8 \\sqrt{3} \\pi ab^2}{27}}\r\n\\]\r\n\u3053\u306e\u3068\u304d, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\nv^2 & = b^2 \\left( 1-\\dfrac{1}{a^2} \\cdot \\dfrac{a^2}{3} \\right) \\\\\r\n& = \\dfrac{2 b^2}{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(C\\) \u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\r\n\\underline{\\dfrac{3 x^2}{a^2} +\\dfrac{3 y^2}{2 b^2} = 1}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b, \u76f4\u7dda \\(\\ell : \\ \\dfrac{x}{a} +\\dfrac{y}{b} = 1\\) \u304c\u3042\u308b. \u305f\u3060\u3057, \\(a , b\\) \u306f\u6b63\u306e\u5b9a\u6570\u3067\u3042\u308b. \u66f2\u7dda \\(C : \\ \\dfrac{x^ … \u7d9a\u304d\u3092\u8aad\u3080