问答题(2022年新高考Ⅰ

已知点A(2,1)在双曲线C:x2/a2 -y2/(a2-1)=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⁡∠PAQ=2√2,求△PAQ的面积.

答案解析

(1)因为点A(2,1)在双曲线C:x2/a2 -y2/(a2-1)=1(a>1)上,所以4/a2 -1/(a2-1)=1,解得 ,即双曲线C:x2/2-y2=1易知直线l的斜率存在,设l:y=kx+m, ,联立可得,(1-2k2 ) x2-4mkx-2m2-2=0,所以,x1+x2=-4mk/(2k2-1),x1 x2=(2m2+2)/(2k2-1),Δ=16m2 k2+4(2m2+2)(2k2-1)>0⇒m2-1+2k2>0.所以由kAP+kBP=0可得,(y2-1)/(x2-2)+(y1-1)/(x1-2)=0,即(x1-2)(kx2+m-1)+(x2-2)(kx1+m-1)=0,即2kx1 x2+(m-1-2k)(x1+x2 )-4(m-1)=0,所以2k×(2m2+2)/(2k2-1)+(m-1-2k)(-4mk/(2k2-1))-4(m-1)=0,化简得,8k2+4k-4+4m(k+1)=0,即(k+1...

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