(本小題滿分11分)已知拋物線
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關(guān)于
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軸對稱,它的頂點(diǎn)在坐標(biāo)原點(diǎn),并且經(jīng)過點(diǎn)
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。
(1)求拋物線的標(biāo)準(zhǔn)方程;
(2)若
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的三個(gè)頂點(diǎn)在拋物線
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上,
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且點(diǎn)
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的橫坐標(biāo)為1,過點(diǎn)
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分別作拋物線
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的切線,兩切線相交于點(diǎn)
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,直線
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與
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軸交于點(diǎn)
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,當(dāng)直線
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的斜率在
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上變化時(shí),直線
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斜率是否存在最大值,若存在,求其最大值和直線
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的方程;若不存在,請說明理由。
(1)
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;(2)略
(1)
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,
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…………………………5分
(2)
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,設(shè)
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,
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,
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設(shè)
BC的斜率為
k,則
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,
又
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,C
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A
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,
直線AC的方程為
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,
令
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……………………6分
AD:
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同理CD:
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,聯(lián)立兩方程得D
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………7分
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………8分
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………9分
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………10分
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………11分
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引兩條切線,切點(diǎn)分別為B,C,且
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為正三角形.
(Ⅰ)求
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最大時(shí)橢圓的方程;
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,過

的直線
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與
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,與橢圓的一個(gè)交點(diǎn)為
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,且
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
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有如下結(jié)論:“圓
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上一點(diǎn)
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處的切線方程為
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”,類比也有結(jié)論:“橢圓
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處的切線方程為
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”,過橢圓C:

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過橢圓
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的右焦點(diǎn),則
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, 滿足條件
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已知直線
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與雙曲線
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。某學(xué)生做了如下變形:由方程組

,消去
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后得到形如

的方程。當(dāng)
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時(shí),該方程有一解,當(dāng)
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時(shí),
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已知橢圓的焦點(diǎn)在
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軸,長軸長
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為10,離心率為
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拋物線的弦與過弦的端點(diǎn)的兩條切線所圍成的三角形常被稱為阿基米德三角形,阿基米德三角形有一些有趣的性質(zhì),如:若拋物線的弦過焦點(diǎn),則過弦的端點(diǎn)的兩條切線的交點(diǎn)在其準(zhǔn)線上.設(shè)拋物線


>
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,弦AB過焦點(diǎn),△ABQ為其阿基米德三角形,則△ABQ的面積的最小值為
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