137-8
(1)c/a=√6/3 a=√(b^2+c^2)=√3 c=a*√6/3=√18/3=√2
b^2=a^2-C^2=3-2=1
椭圆C的方程 x^2/3+y^2=1
(2)设L: y=kx+b 代人x^2/3+y^2=1
x^2+3(kx+b)^2=3
(3k^2+1) x^2+6kbx+3(b^2-1)=0 b^2>4/[3*(1-3k^2)]
x1+x2=-6kb/(3k^2+1)
x1*x2=3(b^2-1)/(3k^2+1)
√
d=!Ax+By+C!/√(A^2+B^2)=!K*0-1*0+b!/√(k^2+1)=√3/2
!d!=√(k^2+1)*√3/2=...全部
(1)c/a=√6/3 a=√(b^2+c^2)=√3 c=a*√6/3=√18/3=√2
b^2=a^2-C^2=3-2=1
椭圆C的方程 x^2/3+y^2=1
(2)设L: y=kx+b 代人x^2/3+y^2=1
x^2+3(kx+b)^2=3
(3k^2+1) x^2+6kbx+3(b^2-1)=0 b^2>4/[3*(1-3k^2)]
x1+x2=-6kb/(3k^2+1)
x1*x2=3(b^2-1)/(3k^2+1)
√
d=!Ax+By+C!/√(A^2+B^2)=!K*0-1*0+b!/√(k^2+1)=√3/2
!d!=√(k^2+1)*√3/2=√[(k^2+1)3/4]
S=1/2!AB!*!d!
!AB!=√[(x2-X1)^2+(y2-y1)^2]
=√[(x2+x1)^2-4x1*x2+k^2(x1+x2)^2-4k^2X1*x2]
=√[(k^2+1)(x1+x2)^2-(k^2+1)(x1+x2)*4]
=√(k^2+1)*√[(x1+x2)^2-4x1x2]
S=1/2*√(k^2+1)*√[(x1+x2)^2-4x1x2]*√[(k^2+1)3/4]
=√3/4(k^2+1))*√[(x1+x2)^2-4x1x2]
==√3/4(k^2+1)/(3k^2+1)*√[12(3k^2-b^2+1)]
=3/2*(k^2+1)/(3k^2+1)*√(3k^2-b^2+1)
=3/4*(k^2+1)*√(9k^2+1)/(3K^2+1)
=3/4*(k^2+1)√[9(k^2+1)-8]/[3(k^2+1)-2]
当k=0
S=3/4最大
。
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