{"id":1924,"date":"2025-07-26T13:35:45","date_gmt":"2025-07-26T04:35:45","guid":{"rendered":"https:\/\/math-friend.com\/?p=1924"},"modified":"2025-08-01T10:03:04","modified_gmt":"2025-08-01T01:03:04","slug":"%e3%80%90-%e4%b9%9d%e5%b7%9e%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%912%e5%86%86%e3%81%ae%e4%ba%a4%e7%82%b9%e3%82%92%e9%80%9a%e3%82%8b%e7%9b%b4%e7%b7%9a%e3%82%92%e4%bd%bf%e3%81%a3%e3%81%a6%e7%82%b9","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=1924","title":{"rendered":"\u3010\u4e5d\u5dde\u5927\u5b66\u5165\u8a66\u3011\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306b\u95a2\u3059\u308b\u6761\u4ef6\u304b\u3089\u30d9\u30af\u30c8\u30eb\u3092\u6c42\u3081\u308b\u554f\u984c(2024)"},"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

\\(xy\\)\u5e73\u9762\u4e0a\u306b2\u70b9\\(\\mathrm{A}\\), \\(\\mathrm{B}\\)\u304c\u3042\u308a, \\(\\mathrm{A}\\)\u306e\u5ea7\u6a19\u306f\\((2,1)\\)\u3067\u3042\u308a, \\(\\mathrm{B}\\)\u306f\u7b2c1\u8c61\u9650\u306b\u3042\u308b. \\(\\mathrm{O}\\)\u3092\u539f\u70b9\u3068\u3057, \\(\\displaystyle |\\overrightarrow{\\mathrm{OB}}|=\\sqrt{10}\\), \\(\\overrightarrow{\\mathrm{OA}}\\perp\\overrightarrow{\\mathrm{AB}}\\)\u304c\u6210\u308a\u7acb\u3063\u3066\u3044\u308b. \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \u70b9\\(\\mathrm{B}\\)\u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.
(2) \\(s\\), \\(t\\)\u3092\u6b63\u306e\u5b9f\u6570\u3068\u3057, \\(\\displaystyle \\overrightarrow{\\mathrm{OC}}=s\\overrightarrow{\\mathrm{OA}}+t \\overrightarrow{\\mathrm{OB}} \\)\u3067\u5b9a\u307e\u308b\u70b9\\(\\mathrm{C}\\)\u306b\u5bfe\u3057\u3066, \u4e09\u89d2\u5f62\\(\\mathrm{OAC}\\)\u306e\u9762\u7a4d\u3068\u4e09\u89d2\u5f62\\(\\mathrm{OBC}\\)\u306e\u9762\u7a4d\u304c\u7b49\u3057\u304f, \\(\\displaystyle |\\overrightarrow{\\mathrm{OC}}|=4\\)\u3067\u3042\u308b\u3068\u304d, \\(s\\), \\(t\\)\u306e\u5024\u3092\u6c42\u3081\u3088.
(2025 \u4e5d\u5dde\u5927\u5b66 \u6587\u7cfb [2])<\/span><\/p>\n\n\n\n

\u3053\u3061\u3089\u306f\u30d9\u30af\u30c8\u30eb\u306e\u57fa\u672c\u554f\u984c\u3067\u3059.

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \\(B\\)\u306e\u5ea7\u6a19\u3092\\((a, b)\\)\u3068\u304a\u304f\u3068, \\(B\\)\u306f\u7b2c1\u8c61\u9650\u306b\u3042\u308b\u304b\u3089, \\(a\\), \\(b\\)\u306f\u5171\u306b\u6b63\u3067\u3042\u308b. \\(\\displaystyle |\\overrightarrow{\\mathrm{OB}}|=\\sqrt{10}\\)\u3088\u308a,
$$
|\\overrightarrow{\\mathrm{OB}}|^2=a^2+b^2=10\\,\\,\\,\u30fb\u30fb\u30fb\u2460
$$\u304c\u6210\u308a\u7acb\u3064\u5fc5\u8981\u304c\u3042\u308b. \u307e\u305f, \\(\\displaystyle \\overrightarrow{\\mathrm{AB}}=(a-2, b-1)\\)\u3067\u3042\u308b\u304b\u3089,
$$
\\begin{align}
\\overrightarrow{\\mathrm{OA}}\\perp\\overrightarrow{\\mathrm{AB}} & \\iff \\overrightarrow{\\mathrm{OA}}\\cdot\\overrightarrow{\\mathrm{AB}}=0\\\\[1.5ex]
& \\iff (2, 1)\\cdot (a-2, b-1)=0 \\\\[1.5ex]
& \\iff 2(a-2)+(b-1)=0\\\\[1.5ex]
& \\iff b=-2a+5\\,\\,\\,\u30fb\u30fb\u30fb\u2461
\\end{align}
$$\u3068\u306a\u308b. \u2461\u3092\u2460\u306b\u4ee3\u5165\u3057\u3066,
$$
\\begin{align}
&a^2+(-2a+5)^2=10 \\\\[1.5ex]
&\\iff 5a^2-20a+15=0\\\\[1.5ex]
&\\iff a^2-4a+3=0\\\\[1.5ex]
&\\iff (a-3)(a-1)=0\\\\[1.5ex]
&\\iff a=1\\,\\,\\, \u307e\u305f\u306f\\,\\,\\, a=3
\\end{align}
$$\u304c\u308f\u304b\u308b. \\(a=3\\)\u306e\u3068\u304d, \\(b=-2\\cdot a+5=-1\\)\u3068\u306a\u308b\u304c, \u3053\u308c\u306f\\(b>0\\) \u306b\u53cd\u3059\u308b\u306e\u3067\u4e0d\u9069\u3067\u3042\u308b. \u3088\u3063\u3066, \\(a=1\\), \\(b=-2\\cdot a+5=3\\)\u304c\u89e3\u3068\u306a\u308a, \\(\\mathrm{B}\\)\u306e\u5ea7\u6a19\u306f\\((1, 3)\\)\u3067\u3042\u308b.<\/p>\n\n\n\n

(2) \\(\\overrightarrow{\\mathrm{OC}}\\)\u3092\\(s\\), \\(t\\)\u3092\u7528\u3044\u3066\u6210\u5206\u8868\u793a\u3059\u308b\u3068,
$$
\\begin{align}
\\overrightarrow{\\mathrm{OC}}&=s\\overrightarrow{\\mathrm{OA}}+t\\overrightarrow{\\mathrm{OB}}\\\\[1.5ex]
&=s(2, 1)+t(1, 3)\\\\[1.5ex]
&=(2s+t, s+3t)
\\end{align}
$$\u3068\u306a\u308b.

\u3053\u308c\u304b\u3089\u4e09\u89d2\u5f62\\(\\mathrm{OAC}\\)\u306e\u9762\u7a4d\u306f,
$$
\\triangle{\\mathrm{OAC}}=\\frac{1}{2}\\left|2(s+3t)-(2s+t)\\right|=\\frac{1}{2}|5t|=\\frac{5t}{2}
$$\u3067\u3042\u308a, \u4e09\u89d2\u5f62\\(\\mathrm{OBC}\\)\u306e\u9762\u7a4d\u306f,
$$
\\triangle{\\mathrm{OBC}}=\\frac{1}{2}\\left|(s+3t)-3(2s+t)\\right|=\\frac{1}{2}|-5s|=\\frac{5s}{2}
$$\u3068\u306a\u308b(\u7d76\u5bfe\u5024\u3092\u5916\u3059\u969b\u306b, \\(s, t>0\\)\u3092\u7528\u3044\u305f). \u3053\u306e2\u3064\u306e\u9762\u7a4d\u304c\u7b49\u3057\u3044\u304b\u3089,
$$
\\frac{5t}{2}=\\frac{5s}{2}\\iff s=t
$$ \u3068\u306a\u308b.

\u3053\u308c\u304b\u3089, \\(\\overrightarrow{\\mathrm{OC}}=(3s, 4s)\\)\u3068\u8868\u305b, \\(\\displaystyle |\\overrightarrow{\\mathrm{OC}}|=4\\)\u3088\u308a,
$$
|\\overrightarrow{\\mathrm{OC}}|=\\sqrt{(3s)^2+(4s)^2}=5s=4
$$\u3067\u3042\u308b\u304b\u3089, \\(\\displaystyle s=\\frac{4}{5}\\)\u3068\u306a\u308a, \\(s=t\\)\u3088\u308a, \\(\\displaystyle s=t=\\frac{4}{5}\\)\u3067\u3042\u308b.
<\/p>\n<\/div><\/div>\n\n\n\n

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