已知點(diǎn)

,

是拋物線

上相異兩點(diǎn),且滿足

.
(Ⅰ)若

的中垂線經(jīng)過點(diǎn)

,求直線

的方程;
(Ⅱ)若

的中垂線交

軸于點(diǎn)

,求

的面積的最大值及此時(shí)直線

的方程.
(Ⅰ)
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(Ⅱ)

.
試題分析:(Ⅰ) 利用導(dǎo)數(shù)分析單調(diào)性,進(jìn)而求最值;(Ⅱ)利用不等式的放縮和數(shù)列的裂項(xiàng)求和
試題解析:(I)方法一
(I)當(dāng)

垂直于

軸時(shí),顯然不符合題意,
所以可設(shè)直線

的方程為

,代入方程

得:

∴

得:

2分
∴直線

的方程為
∵

中點(diǎn)的橫坐標(biāo)為1,∴

中點(diǎn)的坐標(biāo)為

4分
∴

的中垂線方程為
∵

的中垂線經(jīng)過點(diǎn)

,故

,得

6分
∴直線

的方程為

7分
(Ⅱ)由(I)可知

的中垂線方程為

,∴

點(diǎn)的坐標(biāo)為

8分
因?yàn)橹本€

的方程為

∴

到直線

的距離

10分
由

得,

,


12分
∴

, 設(shè)

,則
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,

,

,由

,得

在

上遞增,在

上遞減,當(dāng)

時(shí),

有最大值
得:

時(shí),
直線

方程為

15分
(本題若運(yùn)用基本不等式解決,也同樣給分)
法二:
(Ⅰ)當(dāng)

垂直于

軸時(shí),顯然不符合題意,
當(dāng)

不垂直于

軸時(shí),根據(jù)題意設(shè)

的中點(diǎn)為

,
則

2分
由

、

兩點(diǎn)得

中垂線的斜率為

, 4分
由

,得

6分
∴直線

的方程為

7分
(Ⅱ)由(Ⅰ)知直線

的方程為

8分

中垂線方程為

,中垂線交

軸于點(diǎn)

點(diǎn)

到直線

的距離為

10分
由

得:



當(dāng)

時(shí),

有最大值

,此時(shí)直線

方程為

15分
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科目:高中數(shù)學(xué)
來源:不詳
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已知拋物線的頂點(diǎn)在坐標(biāo)原點(diǎn),焦點(diǎn)在

軸上,且過點(diǎn)

.

(Ⅰ)求拋物線的標(biāo)準(zhǔn)方程;
(Ⅱ)與圓

相切的直線

交拋物線于不同的兩點(diǎn)

若拋物線上一點(diǎn)

滿足


,求

的取值范圍.
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知橢圓

過點(diǎn)

,離心率為

.
(Ⅰ)求橢圓

的方程;
(Ⅱ)過點(diǎn)

且斜率為

(
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)的直線
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與橢圓
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相交于
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兩點(diǎn),直線
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、
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分別交直線
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于
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、
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兩點(diǎn),線段
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的中點(diǎn)為
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.記直線
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的斜率為
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,求證:
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為定值.
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已知?jiǎng)狱c(diǎn)
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到定點(diǎn)
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和
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的距離之和為
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.
(Ⅰ)求動(dòng)點(diǎn)
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軌跡
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的方程;
(Ⅱ)設(shè)
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,過點(diǎn)
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作直線
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,交橢圓
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異于
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的
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兩點(diǎn),直線
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的斜率分別為
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,證明:
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為定值.
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橢圓
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內(nèi)的一點(diǎn)
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,過點(diǎn)P的弦恰好以P為中點(diǎn),那么這弦所在的直線方程( )
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過拋物線
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的焦點(diǎn)
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且傾斜角為
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的直線
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與拋物線在第一、四象限分別交于
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兩點(diǎn),則
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等于( )
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雙曲線

的離心率為( )
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在平面直角坐標(biāo)系
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中,已知
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,
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,
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,
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,其中
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.設(shè)直線
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與
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的交點(diǎn)為
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,求動(dòng)點(diǎn)
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的軌跡的參數(shù)方程(以
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為參數(shù))及普通方程.
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