已知橢圓
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的右準(zhǔn)線
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與
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軸相交于點(diǎn)
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,過橢圓右焦點(diǎn)
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的直線與橢圓相交于
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兩點(diǎn),點(diǎn)
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在右準(zhǔn)線上,且
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軸。
求證:直線
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經(jīng)過線段
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的中點(diǎn)。
由題設(shè),橢圓的半焦距
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,由焦點(diǎn)
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,右準(zhǔn)線方程為
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點(diǎn)
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的坐標(biāo)為
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,
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的中點(diǎn)為
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。
若
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垂直于
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軸,則
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中點(diǎn)為
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,即
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過
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中點(diǎn)
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。
若直線
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不垂直于
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軸,由直線
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過點(diǎn)
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,且由
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軸知點(diǎn)
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不在
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軸上,故直線
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的方程為
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,
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記
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,且
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滿足二次方程
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即
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又
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得
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故直線
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的斜率分別是
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故
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三點(diǎn)共線,所以,直線
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經(jīng)過線段
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的中點(diǎn)
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