{"id":285,"date":"2025-07-03T20:38:55","date_gmt":"2025-07-03T11:38:55","guid":{"rendered":"https:\/\/math-friend.com\/?p=285"},"modified":"2025-08-01T09:24:02","modified_gmt":"2025-08-01T00:24:02","slug":"%e3%80%90%e4%b9%9d%e5%b7%9e%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e5%ba%a7%e6%a8%99%e7%a9%ba%e9%96%93%e5%86%85%e3%81%a7%e5%b9%b3%e9%9d%a2%e3%81%ab%e5%9e%82%e7%9b%b4%e3%81%aa%e7%9b%b4%e7%b7%9a","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=285","title":{"rendered":"\u3010\u4e5d\u5dde\u5927\u5b66\u5165\u8a66\u3011\u5ea7\u6a19\u7a7a\u9593\u5185\u3067\u5e73\u9762\u306b\u5782\u76f4\u306a\u76f4\u7dda\u3068xy\u5e73\u9762\u306e\u4ea4\u70b9\u306b\u95a2\u3059\u308b\u554f\u984c(2025)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\\(\\alpha\\)\u3092\\(xyz\\)\u7a7a\u9593\u5185\u306e3\u70b9\\(A(1,1,-5), B(-1,-1,7), C(1,-1,3)\\)\u3092\u901a\u308b\u5e73\u9762\u3068\u3057. \u70b9\\(P(a,b,t)\\)\u3092\u901a\u308a\\(\\alpha\\)\u306b\u5782\u76f4\u306a\u76f4\u7dda\u3068, \\(xy\\)\u5e73\u9762\u3068\u306e\u4ea4\u70b9\u3092\\(Q\\)\u3068\u3059\u308b.
(1) \u70b9\\(Q\\)\u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.
(2) \\(t\\)\u3092\u5b9f\u6570\u5168\u4f53\u3067\u52d5\u304b\u3059\u3068\u304d, \\(OQ\\)\u306e\u6700\u5c0f\u5024\u304c\\(1\\)\u4ee5\u4e0b\u3068\u306a\u308b\u3088\u3046\u306a\\(a, b\\)\u306e\u6761\u4ef6\u3092\u6c42\u3081\u3088.
(2025 \u4e5d\u5dde\u5927\u5b66\u7406\u7cfb[1])<\/span><\/p>\n\n\n\n

\u5e73\u9762\u306b\u5782\u76f4\u306a\u76f4\u7dda\u306f, \u305d\u306e\u65b9\u5411\u30d9\u30af\u30c8\u30eb\u304c\u5e73\u9762\u4e0a\u306e\\(\\overrightarrow{0}\\)\u3067\u306a\u3044\u4efb\u610f\u306e\u30d9\u30af\u30c8\u30eb\u3068\u76f4\u4ea4\u3057\u307e\u3059. \u3053\u3053\u3067\u306f\u3053\u306e\u6027\u8cea\u3092\u7528\u3044\u3066\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.

\u56f3\u5f62\u554f\u984c\u306f\u56f3\u5f62\u3092\u63cf\u3044\u305f\u65b9\u304c\u3088\u3044\u306e\u3067\u3059\u304c, \u4eca\u56de\u306e\u554f\u984c\u306e\u3088\u3046\u306b3\u6b21\u5143\u306e\u5ea7\u6a19\u7a7a\u9593\u5185\u306b\u5e73\u9762\u3092\u63cf\u304f\u306e\u306f\u96e3\u3057\u3044\u305f\u3081, \u5ea7\u6a19\u306f\u6c17\u306b\u305b\u305a\u3056\u3063\u304f\u308a\u610f\u5473\u304c\u308f\u304b\u308b\u56f3\u3092\u63cf\u3044\u3066\u304a\u304d\u307e\u3059.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\u8981\u306f, \u5e73\u9762\\(\\alpha\\)\u4e0a\u306b\\(A, B, C\\)\u304c\u3042\u3063\u3066, \u7a7a\u9593\u5185\u306e\u70b9\\(P\\)\u304b\u3089\u5e73\u9762\\(\\alpha\\)\u306b\u5782\u7dda\u3092\u4e0b\u308d\u3057\u307e\u3059. \u5e73\u9762\\(\\alpha\\)\u304c\\(yz\\)\u5e73\u9762\u306b\u5bfe\u3057\u3066\u50be\u3044\u3066\u3044\u308b\u305f\u3081, \u305d\u306e\u5782\u7dda\u3092\u5ef6\u9577\u3059\u308b\u3068\u5fc5\u305a\\(xy\\)\u5e73\u9762\u306b\u3076\u3064\u304b\u308a\u307e\u3059. \u305d\u306e\u4ea4\u70b9\u3092\\(Q\\)\u3068\u3057\u3066\u3044\u307e\u3059. \u3053\u306e\u56f3\u3092\u5ff5\u982d\u306b\u7f6e\u3044\u3066, \u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \\(Q\\)\u306f\\(xy\\)\u5e73\u9762\u4e0a\u306e\u70b9\u306a\u306e\u3067, \\((x,y,0)\\)\u3068\u304a\u3051\u308b. \u307e\u305f, \\(\\overrightarrow{PQ}\\)\u306f\u5e73\u9762\\(\\alpha\\)\u306b\u5782\u76f4\u3060\u304b\u3089, \u5e73\u9762\\(\\alpha\\)\u4e0a\u306e\u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{AB}\\), \\(\\overrightarrow{AC}\\)\u3068\u5782\u76f4\u3067\u3042\u308b. \u3053\u308c\u304b\u3089,
$$
\\begin{align}
\\overrightarrow{PQ}\\cdot\\overrightarrow{AB}=0\\\\[1.5ex]
\\overrightarrow{PQ}\\cdot\\overrightarrow{AC}=0
\\end{align}
$$
\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308b.

\u3053\u3053\u3067, \\(\\overrightarrow{PQ}\\), \\(\\overrightarrow{AB}\\), \\(\\overrightarrow{AC}\\)\u3092\u6210\u5206\u8868\u793a\u3059\u308b\u3068,
$$
\\begin{align}
\\overrightarrow{PQ}&=(x,y,0)-(a,b,t)=(x-a,y-b,-t)\\\\[1.5ex]
\\overrightarrow{AB}&=(-1,-1,7)-(1,1,-5)=(-2,-2,12)\\\\[1.5ex]
\\overrightarrow{AC}&=(1,-1,3)-(1,1,-5)=(0,-2,8)
\\end{align}
$$
\u3068\u306a\u308b\u306e\u3067, \u5148\u306e\u5185\u7a4d\\(0\\)\u306e\u6761\u4ef6\u5f0f\u306b\u4ee3\u5165\u3057\u3066,
$$
\\begin{align}
(x-a,y-b,-t)\\cdot(-2,-2,12)&=-2(x-a)-2(y-b)-12t=0\\\\[1.5ex]
(x-a,y-b,-t)\\cdot(0,-2,8)&=-2(y-b)-8t=0
\\end{align}
$$
\u3092\u5f97\u308b. \u3053\u308c\u3092\\(x\\), \\(y\\)\u306b\u3064\u3044\u3066\u89e3\u3051\u3070\u3088\u3044. 2\u756a\u76ee\u306e\u6761\u4ef6\u5f0f\u304b\u3089,
$$
y=b-4t
$$
\u304c\u308f\u304b\u308a, \u3053\u308c\u30921\u756a\u76ee\u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066,
$$
x=a-2t
$$
\u304c\u308f\u304b\u308b.

\u3088\u3063\u3066\\(Q\\)\u306e\u5ea7\u6a19\u306f\\((a-2t,b-4t,0)\\)\u3068\u306a\u308b.<\/p>\n\n\n\n

(2) \\(OQ\\geq 0\\)\u3088\u308a, \\(OQ\\)\u306e\u6700\u5c0f\u5024\u304c\\(1\\)\u4ee5\u4e0b\u3067\u3042\u308b\u3053\u3068\u3068, \\(OQ^2\\)\u306e\u6700\u5c0f\u5024\u304c\\(1\\)\u4ee5\u4e0b\u3067\u3042\u308b\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b.
$$
OQ^2=(a-2t)^2+(b-4t)^2=20t^2-(4a+8b)t+a^2+b^2
$$
\u3068\u306a\u308b. \u3053\u308c\u306f\\(t\\)\u306e2\u6b21\u95a2\u6570\u3068\u306a\u308a, \\(t\\)\u304c\u3059\u3079\u3066\u306e\u5b9f\u6570\u5024\u3092\u3046\u3054\u304f\u3053\u3068\u3068, \\(t^2\\)\u306e\u4fc2\u6570\u304c\\(20\\)\u3067\u6b63\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \u5e73\u65b9\u5b8c\u6210\u3059\u308b\u3053\u3068\u3067\u6700\u5c0f\u5024\u304c\u308f\u304b\u308b.

\u5b9f\u969b\u306b\\(OQ^2\\)\u3092\u5e73\u65b9\u5b8c\u6210\u3059\u308b\u3068,
$$
\\begin{align}
OQ^2&=20t^2-(4a+8b)t+a^2+b^2\\\\[1.5ex]
&=20\\left(t-\\frac{a+2b}{10}\\right)^2+\\frac{(2a-b)^2}{5}
\\end{align}
$$
\u3068\u306a\u308a, \\(t=\\frac{a+2b}{10}\\)\u306e\u3068\u304d, \\(OQ^2\\)\u306f\u6700\u5c0f\u5024\\(\\frac{(2a-b)^2}{5}\\)\u3092\u3068\u308b\u3053\u3068\u304c\u308f\u304b\u308b.

\u3053\u306e\u6700\u5c0f\u5024\u304c\\(1\\)\u4ee5\u4e0b\u3068\u3044\u3046\u3053\u3068\u304b\u3089,
$$
\\frac{(2a-b)^2}{5}\\leq 1 \\iff (2a-b)^2\\leq 5 \\iff -\\sqrt{5}\\leq 2a-b\\leq\\sqrt{5}
$$
\u304c\u6c42\u3081\u308b\u3079\u304d\\(a, b\\)\u306e\u6761\u4ef6\u3067\u3042\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

\u5e73\u65b9\u5b8c\u6210\u3092\u3057\u3066\u3044\u308b\u90e8\u5206\u306f, \\(OQ^2\\)\u3092\\(t\\)\u3067\u5fae\u5206\u3057\u3066, \u6700\u5c0f\u5024\u3092\u3068\u308b\\(t\\)\u3092\u6c42\u3081\u3066\u304b\u3089\\(OQ^2\\)\u306e\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3053\u3068\u3067\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3066\u3082\u69cb\u3044\u307e\u305b\u3093. \u3044\u305a\u308c\u306b\u3057\u3066\u3082\u305d\u308c\u306a\u308a\u306e\u8a08\u7b97\u306b\u306a\u308b\u305f\u3081, \u4eca\u56de\u306f\u5e73\u65b9\u5b8c\u6210\u3067\u6700\u5c0f\u5024\u3092\u5c0e\u51fa\u3057\u307e\u3057\u305f.

\u30b0\u30e9\u30d5\u3092\u66f8\u3051\u3070\u3059\u3050\u308f\u304b\u308b\u3053\u3068\u3067\u3059\u304c, \\(a>0\\)\u306b\u5bfe\u3057\u3066
$$
\\begin{align}
x^2<a &\\iff -\\sqrt{a}<x<\\sqrt{a}\\\\[1.5ex]
x^2>a &\\iff x<-\\sqrt{a} ,\\,\\, x>\\sqrt{a}
\\end{align}
$$
\u306a\u3069\u306f\u3059\u3050\u306b\u4f7f\u3048\u308b\u3088\u3046\u306b\u3057\u3066\u304a\u304d\u307e\u3057\u3087\u3046.

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n