【公式】
【解説】異なる2点(x1 , y1 ),(x2 , y2 )を通る直線の方程式は (1) x1≠x2のとき (2) x1=x2のとき x=x1 高校の数学の教書では,通常,上の公式が書かれています. しかし,数学に苦手意識を持っている生徒に言わせると「xやyが上にも下にもたくさん見えて,目が船酔いのように泳いでしまうので困る」らしい. 実際には,与えられた2点の座標は定数なので,少し見やすくするために文字a,b,c,dで表すと,上の公式は次のようになります.
【公式Ⅱ】
これでx, yが1個ずつになって,直線の方程式らしく見やすくなりましたので,こちらの公式Ⅱの方で解説します.異なる2点(a, b),(c, d)を通る直線の方程式は (1) a≠cのとき (2) a=cのとき x=a ![]() 1点(a, b)を通り,傾きmの直線の方程式は y−b=m(x−a) です.
なぜなら:
傾きmの直線の方程式は傾きy=mx+kと書けますが,この定数項kの値は,点(a, b)を通るということから求めることができ b=ma+k より k=b−ma になります.これを元の方程式に代入すると y=mx+b−ma したがって y−b=m(x−a)…(*1) ![]() 2点(a, b), (c, d)を通る直線の方程式をいきなり考えると,点が2つもあってポイントが絞りきれないので,1点(a, b)を優先的に考える. すなわち,2つ目の点(c, d)は傾きを求めるための材料だけに使う. このとき,2点(a, b), (c, d)を通る直線の傾きは になるから 「2点(a, b), (c, d)を通る直線」は 「1点(a, b)を通り傾き に等しくなる. (*1)により y−b= これで公式Ⅱの(1)が証明された.
この公式において,赤の点線で囲んだ部分は「傾き」を表しているというところがポイントです.
【例】![]() すなわち (2) 2点(−2, 3), (4, −5)を通る直線の方程式は すなわち 次に公式の(2)が x1=x2のとき,なぜ「x=x1」となるのか,「x=x2」ではだめなかのかと考えだしたら分からなくなる場合があります. これは公式Ⅱの(2)でも同様に a=cのとき,なぜ「x=a」となるのか,「x=c」ではだめなかのかというのと同じです. ![]() x=a と言っても x=c といっても,「どちらでもよい」ことになります. 【例】 ![]() x=1 (2) 2点(−2, 3), (−2, 9)を通る直線の方程式は x=−2 |
《問題》 次のうち対応しているものを選びなさい.(右辺は途中までの計算が書いてあります)
○はじめに,左欄の問題文を一つクリックし,続けて右欄の計算式をクリックしたとき,対応していれば消えます.間違えば消えません.
○間違ったときは,右欄を連打するのではなく,左欄を選び直すことから始めてください. |
![]() ![]() |
■[個別の頁からの質問に対する回答][2点を通る直線の方程式について/16.12.22]
xの軸に垂直な直線を引くための方法がわかりません。
=>[作者]:連絡ありがとう.この教材は高校数学Ⅱの教材で「2点を通る直線の方程式」を扱ったものです.あなたが,どのような立場の人で,どのレベルの回答を求めているのか述べていただかないと,答えるのは難しいです. (A) 中学生や高校生が「xの軸に垂直な直線を引く」にはどうすればよいのかと尋ねている場合 この頁のテーマとは全然関係がないので,そういう頁で質問してください. (三角定規の1つの辺をx軸に当てたら,直角になるもう一つの辺がx軸に垂直な直線になります) (B) 点P(a,b)を通りx軸に垂直な直線の方程式は,bの値に関係なく x=a です. (C) 座標変換に対して不変な幾何学的性質の話をしておられるのなら回転移動の一次変換の頁を見て下さい. |
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