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2022年高考数学新高考Ⅰ-21

  2022-12-15 15:54:04  

21.(12分)已知点$A(2,1)$在双曲线$C:\dfrac{x^2}{a^2}-\dfrac{y^2}{{a^2}-1}=1(a > 1)$上,直线$l$交$C$于$P$,$Q$两点,直线$AP$,$AQ$的斜率之和为0.
(1)求$l$的斜率;
(2)若$\tan \angle PAQ=2\sqrt{2}$,求$\Delta PAQ$的面积.
分析:(1)将点$A$代入双曲线方程得$\dfrac{{x}^{2}}{2}-{y}^{2}=1$,由题显然直线$l$的斜率存在,设$l:y=kx+m$,与双曲线联立后,根据直线$AP$,$AQ$的斜率之和为0,求解即可;(2)设直线$AP$的倾斜角为$\alpha$,由$\tan \angle PAQ=2\sqrt{2}$,得$\tan \dfrac{\angle PAQ}{2}=\dfrac{\sqrt{2}}{2}$,联立$\dfrac{y_{1}-1}{x_{1}-2}=\sqrt{2}$,及$\dfrac{x_{1}^{2}}{2}-y_{1}^{2}=1$,根据三角形面积公式即可求解.
解:(1)将点$A$代入双曲线方程得$\dfrac{4}{{a}^{2}}-\dfrac{1}{{a}^{2}-1}=1$,
化简得$a^{4}-4a^{2}+4=0$,$\therefore a^{2}=2$,故双曲线方程为$\dfrac{x^{2}}{2}-y^{2}=1$,
由题显然直线$l$的斜率存在,设$l:y=kx+m$,设$P(x_{1}$,$y_{1})Q(x_{2}$,$y_{2})$,
则联立双曲线得:$(2k^{2}-1)x^{2}+4kmx+2m^{2}+2=0$,
故$x_{1}+x_{2}=-\dfrac{4km}{2k^{2}-1}$,${x}_{1}{x}_{2}=\dfrac{2{m}^{2}+2}{2{k}^{2}-1}$,
${k}_{AP}+{k}_{AQ}=\dfrac{{y}_{1}-1}{{x}_{1}-2}+\dfrac{{y}_{2}-1}{{x}_{2}-2}=\dfrac{k{x}_{1}+m-1}{{x}_{1}-2}+\dfrac{k{x}_{2}+m-1}{{x}_{2}-2}=0$,
化简得:$2kx_{1}x_{2}+(m-1-2k)(x_{1}+x_{2})-4(m-1)=0$,
故$\dfrac{2k(2{m}^{2}+2)}{2{k}^{2}-1}+(m-1-2k)(-\dfrac{4km}{2{k}^{2}-1})-4(m-1)=0$,
即$(k+1)(m+2k-1)=0$,而直线$l$不过$A$点,故$k=-1$;
(2)设直线$AP$的倾斜角为$\alpha$,由$\tan \angle PAQ=2\sqrt{2}$,
$\therefore$$\dfrac{2\tan \dfrac{\angle PAQ}{2}}{1-\tan ^{2}\dfrac{\angle PAQ}{2}}=2\sqrt{2}$,得$\tan \dfrac{\angle PAQ}{2}=\dfrac{\sqrt{2}}{2}$
由$2\alpha +\angle PAQ=\pi$,$\therefore$$\alpha =\dfrac{\pi -\angle PAQ}{2}$,
得$k_{AP}=\tan \alpha =\sqrt{2}$,即$\dfrac{y_{1}-1}{x_{1}-2}=\sqrt{2}$,
联立$\dfrac{y_{1}-1}{x_{1}-2}=\sqrt{2}$,及$\dfrac{x_{1}^{2}}{2}-y_{1}^{2}=1$得$x_{1}=\dfrac{10-4\sqrt{2}}{3},y_{1}=\dfrac{4\sqrt{2}-5}{3}$,
代入直线$l$得$m=\dfrac{5}{3}$,故
$x_{1}+x_{2}=\dfrac{20}{3},x_{1}x_{2}=\dfrac{68}{9}$
而$\vert AP\vert =\sqrt{3}\vert x_{1}-2\vert ,\vert AQ\vert =\sqrt{3}\vert x_{2}-2\vert$,由$\tan \angle PAQ=2\sqrt{2}$,得$\sin \angle PAQ=\dfrac{2\sqrt{2}}{3}$故
$S_{\Delta PAQ}=\dfrac{1}{2}\vert AP\vert \vert AQ\vert \sin \angle PAQ=\sqrt{2}\vert x_{1}x_{2}-2(x_{1}+x_{2})+4\vert =\dfrac{16\sqrt{2}}{9}$.
点评:本题考查了直线与双曲线的综合,属于中档题.

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