如圖所示,射線OA、OB分別與x軸正半軸成45°和30°角,過(guò)點(diǎn)P(1,0)作直線AB分別交OA、OB于A、B兩點(diǎn),當(dāng)AB的中點(diǎn)C恰好落在直線y=
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x上時(shí),求直線AB的方程.
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(3+
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)x-2y-3-
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=0.
解:由題意可得k
OA=tan45°=1,
k
OB=tan(180°-30°)=-
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,
所以射線OA的方程為y=x(x≥0),
射線OB的方程為y=-
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x(x≥0).
設(shè)A(m,m),B(-
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n,n),
所以AB的中點(diǎn)C(
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,
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),
由點(diǎn)C在y=
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x上,且A、P、B三點(diǎn)共線得
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解得m=
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,
所以A(
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,
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).
又P(1,0),
所以k
AB=k
AP=
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=
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,
所以直線AB的方程為y=
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(x-1),
即直線AB的方程為(3+
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)x-2y-3-
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=0.
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、
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是橢圓
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的左、右焦點(diǎn),

為坐標(biāo)原點(diǎn),點(diǎn)

在橢圓上,線段
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與
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滿足
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(Ⅰ)求橢圓的標(biāo)準(zhǔn)方程;
(Ⅱ)過(guò)橢圓的右焦點(diǎn)
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作直線
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已知橢圓

的離心率為
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,直線
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:
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與以原點(diǎn)為圓心、以橢圓

的短半軸長(zhǎng)為半徑的圓相切.
(I)求橢圓
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的方程;
(II)設(shè)橢圓

的左焦點(diǎn)為
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,右焦點(diǎn)
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,直線
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過(guò)點(diǎn)
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且垂直于橢圓的長(zhǎng)軸,動(dòng)直線

垂直
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于點(diǎn)
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,線段
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垂直平分線交
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于點(diǎn)
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