Question

已知椭圆D：y^2/6+x^2/3=1,L是与圆X^2+Y^2=1相切于点C的动直线,L与圆D相交于A,B两点.求iAC

Answer 1

A(x0,y0)B(x1,y1) AB:y=kx+b OC:y=(-1/k)x kx+b=(-1/k)x x=-bk/(1+k^2) y=-b/(1+k^2) x^2+y^2=1 b^2=1+k^2 |AC|^2=x0^2+y0^2-1 |BC|^2=x1^2+y1^2-1 k=(y1-y0)/(x1-x0) b^2=1+k^2 y^2/6+x^2/3=1 a^2=6,b^2=3 BC最大时,|BC|^2=5 B(0,√6) b=√6,k=√5 y-√6=√5x y^2/6+x^2/3=1 y^2+2x^2=6 (√5x+√6)^2+2x^2=6 7x^2+2√30x=0 x(7x+2√30)=0 x=-2√30/7 y=-2√150/7 x0=-2√30/7,y0=-2√150/7 x0^2+y0^2=(120+4*150)/49=720/49 |AC|^2=720/49-1=671/49 |AC|^2/|BC|^2=(671/49)/5=671/245 |AC|/|BC|=√(671/245) BC最小时,|BC|^2=2 B(√3,0) y=kx+b,y=0 x=-b/k=√3 b^2=1+k^2,3k^2=1+k^2 k^2=1/2.k=√2/2 b=-√6/2 y^2+2x^2=6 [(√2/2)x-√6/2]^2+2x^2=6 5x^2/2-√3x-18/4=0 10x^2-4√3x-18=0 10(x-√3/5)^2=96/5 x=-√48/5+√3/5=-2√3/5 y=-√6/5-√6/2=-7√6/10 x0=-2√3/5,y0=-7√6/10 x0^2+y0^2=12/25+49*6/100=(48+294)/100=342/100 |AC|^2=342/100-1=242/100 |AC|^2/|BC|^2=121/100 |AC|/|BC|=11/10 所以 |AC|/|BC|范围：[11/10,√(671/245)]