Question

抛物线过焦点的弦和原点组成的三角形的面积

Answer 1

x^2=2py p>0

焦点F(0,p/2)

过焦点F直线交抛物线于MN

MN直线y=k(x-p/2)

x^2=2pkx-p^2

x^2-2pkx+p^2=0

x1+x2=2pk

x1x2=p^2

MN^2=(x1-x2)^2+(y1-y2)^2=(k^2+1)(x1-x2)^2=(k^2+1)[(x1+x2)^2-4x1x2]

=(k^2+1)[4p^2k^2-4p^2]=4(k^4-1)p^2

MN=2p*√(k^4-1)

O到MN距离d=|k*(-p/2)|/√(1+k^2)

S=(1/2)MN*d=(1/2)p^2√(k^4-1)*|k|/√(1+k^2)=(1/2)p^2|k|√(k^2-1)