こんにちは。この問題は初見で詰まりませんでしたか?定期テストにはよく出てくるかもしれません。それではどうぞ。
【問題】3つの直線,
,
が三角形を作らないとき, 定数
の値を求めよ。
【解答例】
3本の直線が三角形を作らないということは, 以下の3通りの場合が考えられる。
【パターン1】3本の直線が1点で交わるとき(下図イメージ図)
【パターン2】2本の直線
![Rendered by QuickLaTeX.com \ell, n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cefdb70f08d0d4e119307fdacd899ef9_l3.png)
【パターン3】2本の直線
![Rendered by QuickLaTeX.com m, n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f1167b90c6de7ee47693f8b8e91efcee_l3.png)
それでは解いていきましょう。
【パターン1】
直線
![Rendered by QuickLaTeX.com \ell, m](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-fa3e83e9aea8c766fe7846db0958183f_l3.png)
![Rendered by QuickLaTeX.com ( 2, 1 )](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-460a4a59d0b6d4dd32c87910b13a0131_l3.png)
![Rendered by QuickLaTeX.com n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-00037750e74b0d7083c69a4ad2043475_l3.png)
![Rendered by QuickLaTeX.com a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6e4239cd9fe5a53bc98c863c75818b12_l3.png)
よって,
![Rendered by QuickLaTeX.com 2a+(a-1)=-7](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-85ffcbf0f20f0f4bf9deeda7fbbb17a5_l3.png)
![Rendered by QuickLaTeX.com 3a=-6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7a8934e8c88977514b9dc1db48c05c74_l3.png)
![Rendered by QuickLaTeX.com a=-2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-32ca196e84696d52684641f1b46cefad_l3.png)
【パターン2】
直線
![Rendered by QuickLaTeX.com n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-00037750e74b0d7083c69a4ad2043475_l3.png)
![Rendered by QuickLaTeX.com (a-1)y=-ax-7](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-23cda78ff99ab4e026bfe9b5ff8f91ad_l3.png)
![Rendered by QuickLaTeX.com n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-00037750e74b0d7083c69a4ad2043475_l3.png)
![Rendered by QuickLaTeX.com -\dfrac{a}{a-1}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b02ba460fc1de449ed6d7b9bda9dbcaa_l3.png)
これが直線
![Rendered by QuickLaTeX.com \ell](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3179f569c3afe8a05563377e6049016f_l3.png)
![Rendered by QuickLaTeX.com \ell](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3179f569c3afe8a05563377e6049016f_l3.png)
![Rendered by QuickLaTeX.com -2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-eba30760917e2cb7ee25a244fa7ada17_l3.png)
![Rendered by QuickLaTeX.com -\dfrac{a}{a-1}=-2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-896b2601f7cb5df3386574f9d0e0f3fe_l3.png)
![Rendered by QuickLaTeX.com a=2(a-1)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d9186979ab35f085f673b823d8befa5_l3.png)
![Rendered by QuickLaTeX.com a=2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-aba62725c250323076fbd5d71009e978_l3.png)
【パターン3】
パターン2より, 直線
![Rendered by QuickLaTeX.com \ell](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3179f569c3afe8a05563377e6049016f_l3.png)
![Rendered by QuickLaTeX.com -\dfrac{a}{a-1}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b02ba460fc1de449ed6d7b9bda9dbcaa_l3.png)
![Rendered by QuickLaTeX.com m](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-fecc069dce856291a8c07b37e15c926b_l3.png)
したがって,
![Rendered by QuickLaTeX.com -\dfrac{a}{a-1}=1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-493dc3a126c2ca12c528f71a7b6ffc3a_l3.png)
![Rendered by QuickLaTeX.com -a=a-1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-fdf579bd405853d71fefc17db56f4dcc_l3.png)
![Rendered by QuickLaTeX.com a=\dfrac12](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-896b1ec4ed9b364de1a689223c58cd7b_l3.png)
以上より求める定数
![Rendered by QuickLaTeX.com a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6e4239cd9fe5a53bc98c863c75818b12_l3.png)
![Rendered by QuickLaTeX.com a=-2, 2, \dfrac12\cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d35971cb4c35cb9b58df3eb15a4ce4a8_l3.png)