{
  "api_version": "v1",
  "publisher": {
    "name": "Manazoo",
    "url": "https://manazoo.com/"
  },
  "license": "All rights reserved（AI の参照・要約と出典リンクつきの引用は可。転載・再配布・販売は不可）",
  "terms_url": "https://manazoo.com/terms/",
  "disclaimer": "内容の正確さには気をつけていますが、すべてが正しいとは限りません。テストや受験の前には、教科書や先生の説明でも確かめてください。間違いを見つけたら意見箱から知らせてください。",
  "attribution": "事実データの一部は Wikidata（CC0）による。単元の構成は文部科学省「中学校学習指導要領」「高等学校学習指導要領」（政府標準利用規約2.0）。",
  "kind": "quiz",
  "unit": {
    "slug": "plane-curves",
    "title": "平面上の曲線",
    "url": "https://manazoo.com/high/math-c/plane-curves/"
  },
  "canonical_url": "https://manazoo.com/high/math-c/plane-curves/quiz/",
  "questions": [
    {
      "id": "q-ct-tangent-k",
      "type": "input",
      "prompt": "放物線y²＝4xと直線y＝x＋kが接するとき、kの値を求めよ。",
      "choices": null,
      "answer": "1",
      "accept": [],
      "explanation": "x＝y－kを代入しy²－4y＋4k＝0。接する条件はD/4＝4－4k＝0、よってk＝1。",
      "entities": [
        "conic-translation-and-intersection"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-ct-translated-center",
      "type": "input",
      "prompt": "楕円x²/9＋y²/4＝1をx軸方向に2、y軸方向に−1だけ平行移動した楕円の中心のy座標を求めよ。",
      "choices": null,
      "answer": "-1",
      "accept": [],
      "explanation": "元の中心(0,0)をx軸方向に2、y軸方向に−1だけ移すと、新しい中心は(2,−1)。y座標は−1。",
      "entities": [
        "conic-translation-and-intersection"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-ct-translated-equation-choice",
      "type": "choice",
      "prompt": "楕円x²/9＋y²/4＝1をx軸方向に2、y軸方向に−1だけ平行移動した楕円の方程式として正しいものはどれか。",
      "choices": [
        "(x－2)²/4＋(y＋1)²/9＝1",
        "(x＋2)²/9＋(y－1)²/4＝1",
        "(x－2)²/9＋(y＋1)²/4＝1",
        "(x－2)²/9＋(y＋1)²/4＝0"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "xをx－2、yをy－(−1)＝y＋1に置き換えると、(x－2)²/9＋(y＋1)²/4＝1になる。",
      "entities": [
        "conic-translation-and-intersection"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-ct-two-points-condition",
      "type": "choice",
      "prompt": "放物線y²＝4xと直線y＝x＋kの共有点が2個であるためのkの条件として正しいものはどれか。",
      "choices": [
        "k＞0",
        "k＜1",
        "k＜0",
        "k＞1"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "y²－4y＋4k＝0の判別式はD/4＝4－4k。共有点2個はD＞0なので4－4k＞0、整理するとk＜1。",
      "entities": [
        "conic-translation-and-intersection"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-el-c-squared",
      "type": "input",
      "prompt": "楕円x²/25＋y²/16＝1の焦点の座標を(±c,0)とするとき、c²の値を求めよ。",
      "choices": null,
      "answer": "9",
      "accept": [],
      "explanation": "標準形でb²＝a²−c²なので、c²＝a²−b²＝25−16＝9。",
      "entities": [
        "ellipse"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-el-c-value",
      "type": "input",
      "prompt": "楕円x²/25＋y²/16＝1の焦点の座標を(±c,0)とするとき、cの値を求めよ。",
      "choices": null,
      "answer": "3",
      "accept": [],
      "explanation": "c²＝a²−b²＝25−16＝9なので、c＝3。",
      "entities": [
        "ellipse"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-el-standard-choice",
      "type": "choice",
      "prompt": "長軸の長さが10、短軸の長さが8である楕円（焦点がx軸上）の方程式として正しいものはどれか。",
      "choices": [
        "x²/16＋y²/25＝1",
        "x²/5＋y²/4＝1",
        "x²/100＋y²/64＝1",
        "x²/25＋y²/16＝1"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "長軸10より2a＝10でa＝5、短軸8より2b＝8でb＝4。標準形はx²/25＋y²/16＝1。",
      "entities": [
        "ellipse"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-el-sum-distances",
      "type": "input",
      "prompt": "楕円x²/25＋y²/16＝1上の点Pについて、2つの焦点F,F'までの距離PF＋PF'の和を求めよ。",
      "choices": null,
      "answer": "10",
      "accept": [],
      "explanation": "a²＝25よりa＝5。楕円の定義からPF＋PF'＝2a＝10。",
      "entities": [
        "ellipse"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-hy-asymptote-slope",
      "type": "input",
      "prompt": "双曲線x²/9－y²/16＝1の漸近線y＝±(b/a)xのうち、傾きが正の方の傾きの値を求めよ。",
      "choices": null,
      "answer": "4/3",
      "accept": [],
      "explanation": "a²＝9よりa＝3、b²＝16よりb＝4。傾きはb/a＝4/3。",
      "entities": [
        "hyperbola"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-hy-c-value",
      "type": "input",
      "prompt": "双曲線x²/9－y²/16＝1の焦点の座標を(±c,0)とするとき、cの値を求めよ。",
      "choices": null,
      "answer": "5",
      "accept": [],
      "explanation": "標準形でb²＝c²−a²なので、c²＝a²＋b²＝9＋16＝25、c＝5。",
      "entities": [
        "hyperbola"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-hy-pf-diff",
      "type": "input",
      "prompt": "双曲線x²/9－y²/16＝1上の点Pについて、2つの焦点からの距離の差|PF－PF'|を求めよ。",
      "choices": null,
      "answer": "6",
      "accept": [],
      "explanation": "a²＝9よりa＝3。双曲線の定義から|PF－PF'|＝2a＝6。",
      "entities": [
        "hyperbola"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-hy-vertex-choice",
      "type": "choice",
      "prompt": "双曲線x²/9－y²/16＝1の頂点の座標として正しいものはどれか。",
      "choices": [
        "(±4,0)",
        "(±9,0)",
        "(±5,0)",
        "(±3,0)"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "標準形x²/a²－y²/b²＝1の頂点は(±a,0)。a²＝9よりa＝3なので頂点は(±3,0)。",
      "entities": [
        "hyperbola"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-pb-directrix-x",
      "type": "input",
      "prompt": "放物線y²＝8xの準線の方程式はx＝aの形である。aの値を求めよ（符号も含む）。",
      "choices": null,
      "answer": "-2",
      "accept": [],
      "explanation": "4p＝8よりp＝2。準線はx＝−pなのでx＝−2。",
      "entities": [
        "parabola"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-pb-focus-x",
      "type": "input",
      "prompt": "放物線y²＝8xの焦点Fの座標を(a,0)の形で表すとき、aの値を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "標準形y²＝4pxとy²＝8xを比べると4p＝8、よってp＝2。焦点はF(p,0)＝(2,0)。",
      "entities": [
        "parabola"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-pb-pf-distance",
      "type": "input",
      "prompt": "放物線y²＝8x上の点P(2,4)について、焦点Fとの距離PFを求めよ。",
      "choices": null,
      "answer": "4",
      "accept": [],
      "explanation": "放物線ではPFは準線までの距離PHと等しく、PH＝（Pのx座標）－（準線のx座標）＝2－(−2)＝4。",
      "entities": [
        "parabola"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-pb-standard-choice",
      "type": "choice",
      "prompt": "焦点が(3,0)、準線がx＝−3である放物線の方程式として正しいものはどれか。",
      "choices": [
        "y²＝12x",
        "x²＝12y",
        "y²＝6x",
        "y²＝3x"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "焦点F(p,0)＝(3,0)なのでp＝3。標準形y²＝4pxに当てはめるとy²＝12x。",
      "entities": [
        "parabola"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-pc-cycloid-x-formula-choice",
      "type": "choice",
      "prompt": "半径aの円が直線上をすべらずにθだけ回転したとき、円の中心はx軸方向にどれだけ移動するか。",
      "choices": [
        "aθ²",
        "aθ",
        "a",
        "2aθ"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "円がすべらずθだけ回転すると、転がった弧の長さaθの分だけ中心もx軸方向に進む。",
      "entities": [
        "parametric-curves"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-pc-cycloid-y-range",
      "type": "input",
      "prompt": "サイクロイド曲線x＝a(θ－sinθ)、y＝a(1－cosθ)で、a＝1のときθ＝πにおけるyの値を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "cosπ＝－1なので、y＝1×(1－(－1))＝2。",
      "entities": [
        "parametric-curves"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-pc-eliminate-t",
      "type": "input",
      "prompt": "媒介変数tを用いてx＝2t、y＝t²＋1と表される曲線は、tを消去するとy＝ax²＋bの形になる。aの値を求めよ。",
      "choices": null,
      "answer": "1/4",
      "accept": [],
      "explanation": "x＝2tよりt＝x/2。y＝t²＋1に代入するとy＝(x/2)²＋1＝(1/4)x²＋1になり、a＝1/4。",
      "entities": [
        "parametric-curves"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-pc-point-at-t",
      "type": "input",
      "prompt": "x＝2t、y＝t²＋1について、t＝3のときのxの値を求めよ。",
      "choices": null,
      "answer": "6",
      "accept": [],
      "explanation": "x＝2tにt＝3を代入するとx＝2×3＝6。",
      "entities": [
        "parametric-curves"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-po-angle-radian-choice",
      "type": "choice",
      "prompt": "極座標の偏角θを表す弧度法について、90°はラジアンでどれか。",
      "choices": [
        "2π",
        "π",
        "π/4",
        "π/2"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "180°＝πラジアンなので、90°はその半分のπ/2ラジアン。",
      "entities": [
        "polar-coordinates"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-po-circle-coefficient",
      "type": "input",
      "prompt": "円(x－2)²＋y²＝4を極方程式r＝a cosθの形で表すとき、aの値を求めよ。",
      "choices": null,
      "answer": "4",
      "accept": [],
      "explanation": "中心(2,0)、半径2の円の極方程式はr＝2×2×cosθ＝4cosθになり、r＝a cosθのaは4。",
      "entities": [
        "polar-coordinates"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-po-line-polar-form",
      "type": "input",
      "prompt": "直線x＝3を極方程式で表すとr cosθ＝aの形になる。aの値を求めよ。",
      "choices": null,
      "answer": "3",
      "accept": [],
      "explanation": "x＝r cosθなので、x＝3はそのままr cosθ＝3になる。",
      "entities": [
        "polar-coordinates"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-po-rect-x",
      "type": "input",
      "prompt": "極座標(r,θ)＝(4,60°)の点を直交座標(x,y)で表すとき、xの値を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "x＝r cosθ＝4×cos60°＝4×(1/2)＝2。",
      "entities": [
        "polar-coordinates"
      ],
      "verified": true,
      "placeholder": false
    }
  ]
}