\\(t\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b.\r\n\u539f\u70b9\u3092 O \u3068\u3059\u308b\u7a7a\u9593\u5185\u306b, \\(2\\) \u70b9 A \\(( 2t , 2t , 0 )\\) , B \\(( 0 , 0 , t )\\) \u304c\u3042\u308b. \u307e\u305f\u52d5\u70b9 P \u306f\r\n\\[\r\n\\overrightarrow{\\text{OP}} \\cdot \\overrightarrow{\\text{AP}} +\\overrightarrow{\\text{OP}} \\cdot \\overrightarrow{\\text{BP}} +\\overrightarrow{\\text{AP}} \\cdot \\overrightarrow{\\text{BP}} = 3\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u52d5\u304f. OP \u306e\u6700\u5927\u5024\u304c \\(3\\) \u3068\u306a\u308b\u3088\u3046\u306a \\(t\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\(\\overrightarrow{a} = \\overrightarrow{\\text{OA}}\\) , \\(\\overrightarrow{b} = \\overrightarrow{\\text{OB}}\\) , \\(\\overrightarrow{p} = \\overrightarrow{\\text{OP}}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\overrightarrow{a} \\cdot \\overrightarrow{b} & = 0 \\\\\r\n\\left| \\overrightarrow{a} +\\overrightarrow{b} \\right| & = t \\sqrt{2^2 +2^2 +1} = 3t\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u308c\u3070, \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n\\overrightarrow{p} \\cdot \\left( \\overrightarrow{p} -\\overrightarrow{a} \\right) & +\\overrightarrow{p} \\cdot \\left( \\overrightarrow{p} -\\overrightarrow{b} \\right) +\\left( \\overrightarrow{p} -\\overrightarrow{a} \\right) \\cdot \\left( \\overrightarrow{p} -\\overrightarrow{b} \\right) = 3 \\\\\r\n3 \\left| \\overrightarrow{p} \\right|^2 & -2 \\left( \\overrightarrow{a} +\\overrightarrow{b} \\right) \\cdot \\overrightarrow{p} +\\overrightarrow{a} \\cdot \\overrightarrow{b} = 3 \\\\\r\n& \\left| \\overrightarrow{p} \\right|^2 -2 \\dfrac{\\overrightarrow{a} +\\overrightarrow{b}}{3} \\cdot \\overrightarrow{p} = 1 \\\\\r\n& \\left| \\overrightarrow{p} -\\dfrac{\\overrightarrow{a} +\\overrightarrow{b}}{3} \\right|^2 = \\left| \\dfrac{\\overrightarrow{a} +\\overrightarrow{b}}{3} \\right|^2 +1 \\\\\r\n\\text{\u2234} \\quad & \\left| \\overrightarrow{p} -\\dfrac{\\overrightarrow{a} +\\overrightarrow{b}}{3} \\right|^2 = t^2 +1\r\n\\end{align}\\]\r\n\u25b3OAB \u306e\u91cd\u5fc3\u3092 G \\(\\left( \\dfrac{2t}{3} , \\dfrac{2t}{3} , \\dfrac{t}{3} \\right)\\) \u3068\u304a\u3051\u3070, \u70b9 P \u306f\u70b9 G \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(\\sqrt{t^2+1}\\) \u306e\u7403\u9762\u4e0a\u3092\u52d5\u304f.
\r\n\\(\\text{OG} = t\\) \u306a\u306e\u3067, OP \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\nt +\\sqrt{t^2+1}\r\n\\]\r\n\u3088\u3063\u3066, \u6761\u4ef6\u3092\u307f\u305f\u3059 \\(t\\) \u306e\u5024\u306f\r\n\\[\\begin{align}\r\nt +\\sqrt{t^2+1} & = 3 \\\\\r\nt^2 +1 & = (3-t)^2 , \\ 3-t \\gt 0 \\\\\r\n6t & = 8 , \\ t \\lt 3 \\\\\r\n\\text{\u2234} \\quad t & = \\underline{\\dfrac{4}{3}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(t\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b. \u539f\u70b9\u3092 O \u3068\u3059\u308b\u7a7a\u9593\u5185\u306b, \\(2\\) \u70b9 A \\(( 2t , 2t , 0 )\\) , B \\(( 0 , 0 , t )\\) \u304c\u3042\u308b. \u307e\u305f\u52d5\u70b9 P \u306f \\[ \\overright … \u7d9a\u304d\u3092\u8aad\u3080