顶点在原点,焦点在x轴的抛物线截直线y=-2x-1所得的弦长|AB|=5√3,求抛物线的方程

直线y=-2x-1过2,3,4象限,焦点在x轴与直线相交
∴抛物线在2,3象限 ∴焦点(-p/2,0) ,
抛物线过原点,设方程 ：y^2=-2px (p>0)
将直线代入抛物线方程：(-2x-1)^2=4x^2+4x+1=-2px 4x^2+(4+2p)x+1=0
x1+x2=-(4+2p)/4 ,x1x2=1/4
|AB|^2=(x1-x2)^2+(y1-y2)^2=(5√3)^2
=(x1-x2)^2+4(x1-x2)^2
=5(x1-x2)^2
=5[(x1+x2)^2-4x1x2]
=5[(-(4+2p)/4)^2-4*1/4]
=5[(p+2)^2-1]
=25*3
=> (p+2)^2-1=15 解得p=2 (p=-6舍弃)
∴抛物线方程为：y^2=-2px=-4x