こんにちは。今回は2022年の徳島県の公立高校の入試問題から放物線の問題をやってみようと思います。
【問題】下の図のように, 関数のグラフ上に2点A, Bがあり, 点Aの
座標は
, 点Bの座標は2である。また, 直線ABと
軸との交点をCとする。(1)~(3)に答えなさい。
(1) 点Aの
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
(2)
![Rendered by QuickLaTeX.com a=\dfrac12](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-896b1ec4ed9b364de1a689223c58cd7b_l3.png)
(3)
![Rendered by QuickLaTeX.com a=1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4650e0afe8d05e59dc367be97f5269ce_l3.png)
(a) △OABの面積を求めなさい。
(b) 線分ACの中点をPとし, 点Qを関数
![Rendered by QuickLaTeX.com y=ax^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3d90f33a3d8d1a392fac89732ce270f4_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
「2022年徳島県公立高校入試」
(1)点Aの座標が6のとき, Aの座標は
である。これを原点Oを中心に180°回転(点対称移動)させることは, 点Aを原点について対称な座標を求めることと等しいので,
座標,
座標の符号を変えればよい。したがって, 答えは,
(答)
(2) のとき, 放物線の関数は
となり, 点A, Bの座標はそれぞれ, A
, B
となるので, 求める長さは,
(答)
(3)
(a) のとき, 放物線の関数は
となり, 点A, Bの座標はそれぞれ, A
, B
となり, この2点から直線ABの式は
と分かる。
求める三角形は
![Rendered by QuickLaTeX.com 8\times6\times\dfrac12=24\cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b7ac391c62252f65a0765aed79b1778e_l3.png)
(b) 方針としては, Pの座標を求め, Qの
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
図のように, Pの座標はA
![Rendered by QuickLaTeX.com (-4, 16)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c5524b621dbd3d5c568bbfff7dcbb5c1_l3.png)
![Rendered by QuickLaTeX.com (0, 8)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-30fb28839472a1a7feea5cafb0339b1e_l3.png)
![Rendered by QuickLaTeX.com (-2, 12)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f8ee8f3a7278217c4d183961e84ef72c_l3.png)
Qの
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b406bbb54258d4eecd44d6c4f4f19e0f_l3.png)
![Rendered by QuickLaTeX.com (t, t^2)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2030eb2d887c9d0488a5e41cfac0beb2_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com 12, t^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-875fc0077e7c8d07c7cae675cc6ee9a7_l3.png)
![Rendered by QuickLaTeX.com t+2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-32c4f8379f45122ee2eb887b6534d539_l3.png)
![Rendered by QuickLaTeX.com (t^2+12)\times(t+2)\times\dfrac12=\dfrac12(t^2+12)(t+2)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-bef7c5769cc2b9693195b16a5bbd7e18_l3.png)
求める△OPQはこの台形PQSRの面積から△PROと△QSOを引いたものなので,
△PROと△QSOの面積をそれぞれ求めると,
△PRO
![Rendered by QuickLaTeX.com =2\times12\times\dfrac12=12](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-839d7c46a4f82a7207e36055c65fde52_l3.png)
△QSO
![Rendered by QuickLaTeX.com =t\times t^2\times\dfrac12=\dfrac12 t^3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c3b809c4da1fe585a8bc0b20e9320eae_l3.png)
よって, △OPQは
△OPQ
![Rendered by QuickLaTeX.com =\dfrac12(t^2+12)(t+2)-12-\dfrac12 t^3=t^2+6t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2ccc1bf56de61a4963a67853bf874a3d_l3.png)
これが24(△OABの面積)になればいいので,
![Rendered by QuickLaTeX.com t^2+6t=24](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d2346792c689df667196dfc136e96a76_l3.png)
![Rendered by QuickLaTeX.com t^2+6t-24=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-456c68ad1d8c9e26dea2562da45ebed1_l3.png)
これを解いて,
![Rendered by QuickLaTeX.com t=-3\pm\sqrt{33}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3bd98da5d38e93a0024f242c360175bf_l3.png)
![Rendered by QuickLaTeX.com t>0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-1ac713d88ec484de8d54c865d95f7a49_l3.png)
![Rendered by QuickLaTeX.com t=-3+\sqrt{33}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d30aa539f21dd20dac0525a95c64ca7a_l3.png)
よって, 求める座標は
![Rendered by QuickLaTeX.com -3+\sqrt{33}\cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9b178353e4d4acfc84cbdc0ff279b755_l3.png)