直线l与抛物线y∧2=2px交于a(x1,y1),b(x2,y2)两点,若y1y2=-p∧2,求证:直线l过抛物线的焦点f
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![直线l与抛物线y∧2=2px交于a(x1,y1),b(x2,y2)两点,若y1y2=-p∧2,求证:直线l过抛物线的焦点f](/uploads/image/z/13879860-60-0.jpg?t=%E7%9B%B4%E7%BA%BFl%E4%B8%8E%E6%8A%9B%E7%89%A9%E7%BA%BFy%E2%88%A72%EF%BC%9D2px%E4%BA%A4%E4%BA%8Ea%28x1%2Cy1%29%2Cb%28x2%2Cy2%29%E4%B8%A4%E7%82%B9%2C%E8%8B%A5y1y2%EF%BC%9D-p%E2%88%A72%2C%E6%B1%82%E8%AF%81%EF%BC%9A%E7%9B%B4%E7%BA%BFl%E8%BF%87%E6%8A%9B%E7%89%A9%E7%BA%BF%E7%9A%84%E7%84%A6%E7%82%B9f)
直线l与抛物线y∧2=2px交于a(x1,y1),b(x2,y2)两点,若y1y2=-p∧2,求证:直线l过抛物线的焦点f
直线l与抛物线y∧2=2px交于a(x1,y1),b(x2,y2)两点,若y1y2=-p∧2,求证:直线l过抛物线的焦点f
直线l与抛物线y∧2=2px交于a(x1,y1),b(x2,y2)两点,若y1y2=-p∧2,求证:直线l过抛物线的焦点f
设直线l的方程为y = kx + b,x = (y - b)/k
代入y² = 2px:
y² = 2p(y-b)/k
ky² - 2py + 2pb = 0
y1*y2 = 2pb/k = -p²
b = -pk/2
l的方程为y = kx -pk/2 = k(x - p/2)
显然,l过点(p/2,0),此外为该抛物线的焦点.