文数：过抛物线y^2=2px的焦点F作倾斜角为135°的直线,交抛物线与A,B两点过抛物线y^2=2px(p>0)的焦点F作倾斜角为135°的直线,交抛物线于A、B两点,则三角形OAB的面积A.√2/2p^2B.√2p^2C.p^2

Question

文数：过抛物线y^2=2px的焦点F作倾斜角为135°的直线,交抛物线与A,B两点
过抛物线y^2=2px(p>0) 的焦点F作倾斜角为135°的直线,交抛物线于A、B两点,则三角形OAB的面积 A.√2/2p^2
B.√2p^2
C.p^2

Answer 1

抛物线y^2=2px(p>0) 的焦点F（p/2,0）
k=-1
直线方程为
y=-x+p/2 x=-y+p/2
y^2=2px
消x得
y^2=-2py+p^2
整理得 y^2+2py-p^2=0
y1y2=-p^2 y1+y2=-2p
|y1-y2|=根号[(y1+y2)^2-4y1y2]
=根号[4p^2+4p^2]
=4根号2p
S△OAB=S△OFA+S△OFB
=1/2*|y1-y2|*OF
=1/2*4根号2p*p/2
=根号2p^2
选 B