Question

已知抛物线L的方程为x^=2py(p>0),直线y=x截抛物线L所的弦长为根号2,求p的值

Answer 1

设直线与抛物线交于(x1,y1),(x2,y2)两点.

y=x代入抛物线方程,整理,得x^2-2px=0,x1、x2是方程的两根.

x(x-2p)=0

x1=0 x2=2p

y1=0 y2=2p

弦长=√[(x1-x2)^2+(y1-y2)^2]

=√[(2p-0)^2+(2p-0)^2]

=2√2|p|=√2

|p|=1/2

p=1/2或p=-1/2.