こんにちは。今回は大阪府のC問題を取り上げてみます。いい問題ですので取り組んでみてください。それではどうぞ。
,
を正の定数とする。下の図において,
は関数
のグラフを表し,
は関数
のグラフを表す。
は
と平行な直線であり, その切片は
である。四角形ABCDは正方形であり, ABは
軸に平行であって, 辺ADは
軸に平行である。Aは
上にあり, その
座標は4である。Bは
上にあり, Dは
上にある。Cの
座標は
であり, Cの
座標はBの
座標より小さい。
,
の値をそれぞれ求めなさい。途中の式も含めた求め方も書くこと。ただし, 座標軸の1めもりの長さは1cmであるとする。
![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/02/1yohaku.png)
答え
問題より, 正方形の一辺の長さは6とわかります。![Rendered by QuickLaTeX.com 4-(-2)=6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4bff0f5a0ae56279c91179c36d5c8053_l3.png)
また,
,
は平行なので,
の式は
と置くことができます。
A, B, C, Dの座標を
,
を用いて表すと,
A(
,
), B(
,
), C(
,
), D(
,
)
※A(
,
)としても可
このとき, 辺BCの長さを
を用いて表すと,
![Rendered by QuickLaTeX.com (-2b+4)-(4b-3)=-6b+7](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c5853d8cf1eeaaec4e5662f32d1aa336_l3.png)
これが6と等しいので,
![Rendered by QuickLaTeX.com -6b+7=6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-02e7c1003fb4abdf8aeb63360294aebc_l3.png)
![Rendered by QuickLaTeX.com b=\dfrac16](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-41e55ea927e9dbf89d0614f48262476a_l3.png)
またA, Bの
座標は等しいので,
![Rendered by QuickLaTeX.com 16a=-2b+4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-71a346c62cdc540afeb9ea3906b9e2f2_l3.png)
これに
を代入し,
![Rendered by QuickLaTeX.com a=\dfrac{11}{48}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d9b67c9ac71c042b860dfa0df04f5904_l3.png)
よって
・・・答え
![Rendered by QuickLaTeX.com 4-(-2)=6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4bff0f5a0ae56279c91179c36d5c8053_l3.png)
また,
![Rendered by QuickLaTeX.com \ell](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3179f569c3afe8a05563377e6049016f_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 n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-00037750e74b0d7083c69a4ad2043475_l3.png)
![Rendered by QuickLaTeX.com y=bx-3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7486e0bafb64943c1838ef0e3afb3bb2_l3.png)
A, B, C, Dの座標を
![Rendered by QuickLaTeX.com a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6e4239cd9fe5a53bc98c863c75818b12_l3.png)
![Rendered by QuickLaTeX.com b](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7d3b74b93cf8d3562b537c81c243a48c_l3.png)
A(
![Rendered by QuickLaTeX.com 4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d9d67cb85eca77cdbabd48cabd8368a0_l3.png)
![Rendered by QuickLaTeX.com 16a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e4f83ff9e499066dd5a3c10eaddb5497_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 -2b+4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-08c9262168dac44b22875956308ba0d4_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 4b-3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3b81821a60c651778b2c5e221da330c7_l3.png)
![Rendered by QuickLaTeX.com 4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d9d67cb85eca77cdbabd48cabd8368a0_l3.png)
![Rendered by QuickLaTeX.com 4b-3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3b81821a60c651778b2c5e221da330c7_l3.png)
※A(
![Rendered by QuickLaTeX.com 4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d9d67cb85eca77cdbabd48cabd8368a0_l3.png)
![Rendered by QuickLaTeX.com -2b+4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-08c9262168dac44b22875956308ba0d4_l3.png)
このとき, 辺BCの長さを
![Rendered by QuickLaTeX.com b](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7d3b74b93cf8d3562b537c81c243a48c_l3.png)
![Rendered by QuickLaTeX.com (-2b+4)-(4b-3)=-6b+7](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c5853d8cf1eeaaec4e5662f32d1aa336_l3.png)
これが6と等しいので,
![Rendered by QuickLaTeX.com -6b+7=6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-02e7c1003fb4abdf8aeb63360294aebc_l3.png)
![Rendered by QuickLaTeX.com b=\dfrac16](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-41e55ea927e9dbf89d0614f48262476a_l3.png)
またA, Bの
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com 16a=-2b+4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-71a346c62cdc540afeb9ea3906b9e2f2_l3.png)
これに
![Rendered by QuickLaTeX.com b=\dfrac16](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-41e55ea927e9dbf89d0614f48262476a_l3.png)
![Rendered by QuickLaTeX.com a=\dfrac{11}{48}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d9b67c9ac71c042b860dfa0df04f5904_l3.png)
よって
![Rendered by QuickLaTeX.com a=\dfrac{11}{48}, b=\dfrac16](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2b57fef88012d7c00b7dcf07a37510bd_l3.png)