| 2025年高考数学新高考Ⅱ-6 |
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2026-10-10 17:44:40 |
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(5分)设抛物线$C:y^{2}=2px(p > 0)$的焦点为$F$,点$A$在$C$上,过$A$作$C$准线的垂线,垂足为$B$.若直线$BF$的方程为$y=-2x+2$,则$\vert AF\vert =$( ) A.3 B.4 C.5 D.6 〖答案〗$C$ 〖分析〗写出抛物线的焦点和准线,设$A(x_{0}$,$y_{0})$,得$B(-\dfrac{p}{2},{y}_{0})$,由点$B$、$F$在直线上建立方程,求出$p$和$y_{0}$,再由点$A$在$C$上求出$x_{0}$,再由焦半径公式即可求得. 〖解答〗解:由题知,$F(\dfrac{p}{2}$,$0)$,准线方程为:$x=-\dfrac{p}{2}$, 设$A(x_{0}$,$y_{0})$,则$B(-\dfrac{p}{2},{y}_{0})$, 因为$l_{BF}:y=-2x+2$, 所以$\left\{\begin{array}{l}{{y}_{0}=-2\times (-\dfrac{p}{2})+2}\\ {0=-2\times \dfrac{p}{2}+2}\end{array}\right.$,解得$\left\{\begin{array}{l}{p=2}\\ {{y}_{0}=4}\end{array}\right.$, 因为点$A$在$C$上,所以${{y}_{0}}^{2}=2p{x}_{0}$,即$16=4x_{0}$,所以$x_{0}=4$, 所以$\vert AF\vert ={x}_{0}+\dfrac{p}{2}=4+1=5$. 故选:$C$. 〖点评〗本题考查抛物线的定义和直线方程的应用,属于基础题.
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