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2024年高考数学甲卷-文21

  2024-08-28 22:20:27  

(12分)已知椭圆$C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1(a > b > 0)$的右焦点为$F$,点$M(1,\dfrac{3}{2})$在椭圆$C$上,且$MF\bot x$轴.
(1)求椭圆$C$的方程;
(2)过点$P(4,0)$的直线与椭圆$C$交于$A$,$B$两点,$N$为线段$FP$的中点,直线$NB$与$MF$交于$Q$,证明:$AQ\bot y$轴.

分析:(1)根据已知条件,结合椭圆的定义,以及勾股定理,求出$a$,再结合椭圆的性质,求出$b$,即可求解;
(2)结合向量的坐标运算,推得$\left\{\begin{array}{l}{\lambda {x}_{2}=4+4\lambda -{x}_{1}}\\ {\lambda {y}_{2}=-{y}_{1}}\end{array}\right.$,再结合点$A$,$B$两点位于椭圆上,求出等式,再结合直线$NB$与$MF$交于$Q$,即可求解.
解:(1)设椭圆$C$的左焦点为$F_{1}$,
点$M(1,\dfrac{3}{2})$在椭圆$C$上,且$MF\bot x$轴,
则$\vert F_{1}F\vert =2$,$\vert MF\vert =\dfrac{3}{2}$,
由勾股定理可知,$\vert M{F}_{1}\vert =\dfrac{5}{2}$,
故$2a=\vert MF_{1}\vert +\vert MF\vert =4$,解得$a^{2}=4$,$b^{2}=a^{2}-1=3$,
故椭圆$C$的方程为$\dfrac{x^2}{4}+\dfrac{y^2}{3}=1$;
(2)证明:设$A(x_{1}$,$y_{1})$,$B(x_{2}$,$y_{2})$,
$\overrightarrow{AP}=\lambda \overrightarrow{PB}$,
则$\left\{\begin{array}{l}{\dfrac{{x}_{1}+\lambda {x}_{2}}{1+\lambda }=4}\\ {\dfrac{{y}_{1}+\lambda {y}_{2}}{1+\lambda }=0}\end{array}\right.$,即$\left\{\begin{array}{l}{\lambda {x}_{2}=4+4\lambda -{x}_{1}}\\ {\lambda {y}_{2}=-{y}_{1}}\end{array}\right.$①,
又由$\left\{\begin{array}{l}{3{x}_{1}^{2}+4{y}_{1}^{2}=12}\\ {3(\lambda {x}_{2})^{2}+4(\lambda {y}_{2})^{2}=12{\lambda }^{2}}\end{array}\right.$可得,$3\cdot \dfrac{x_1+\lambda x_2}{1+\lambda }\cdot \dfrac{x_1-\lambda x_2}{1-\lambda }+4\dfrac{y_1+\lambda y_2}{1+\lambda }\cdot \dfrac{y_1-\lambda y_2}{1-\lambda }=12$②,
结合①②可得,$5\lambda -2\lambda x_{2}+3=0$,
$P(4,0)$,$F(1,0)$,$N(\dfrac{5}{2},0)$,$B(x_{2}$,$y_{2})$,
则直线$NB$的方程为$y-0=\dfrac{{y}_{2}}{{x}_{2}-\dfrac{5}{2}}(x-\dfrac{5}{2})$,
$MF\bot x$轴,直线$NB$与$MF$交于$Q$,
则$x_{Q}=1$,
故$y_Q=\dfrac{3y_2}{5-2x_2}=\dfrac{3\lambda y_2}{5\lambda -2\lambda x_2}=-\lambda y_2=y_1$,
故$AQ\bot y$轴.
点评:本题主要考查直线与椭圆的综合,考查转化能力,属于难题.

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