直线与圆锥曲线

x^2/16+y^2/4=1,
X^2+4Y^2-16=0,y=kx+b,
X^2+4(kx+b)^2-16=0,
(1+4k^2)x^2+8kbx+4b^2-16=0,
x1+x2=-8kb/(1+4k^2),
x1*x2=(4b^2-16)/(1+4k^2),

(x2-x1)^2=(x1+x2)^2-4x1*x2=[(-8kb)/(1+4k^2)]^2-4*(4b^2-16)/(1+4k^2),
而,K=(Y2-Y1)/(X2-X1),则有,
(y2-y1)^2=k^2*(x2-x1)=k^2*{[(-8kb)/(1+4k^2)]^2-4*(4b^2-16)/(1+4k^2)},

令,点O到AB的距离为:d,
y=kx+b,
d=h=|b|/√(1+k^2),
S=4=1/2*|AB|*d,则有
|AB|=4=8/d,d=2=|b|/√(1+k^2),
b^2=4(1+k^2),则有,
(x2-x1)^2=64*3k^2/(1+4k^2)^2,
(y2-y1)^2=64*3k^4/(1+4k^2)^2.
而,|AB|^2=16=(X2-X1)^2+(Y2-Y1)^2,
16=64*3k^2/(1+4k^2)^2+64*3k^4/(1+4k^2)^2.
(1+4k^2)^2=12k^2+12k^4,
(4k^4-4k^2+1=0,
(2k^2-1)^2=0, k^2=1/2.
k=±√2/2,
k1=√2/2,k2=-√2/2.
b^2=4(k^2+1)=6,
b1=√6,b2=-√6.

直线AB的方程的方程为:
y=√2/2*x+√6或Y=-√2/2*X-√6.