焦點(diǎn)在x軸上,設(shè)拋物線方程為y² = 2px,焦點(diǎn)(p/2,0)
設(shè)A點(diǎn)坐標(biāo)(x1,y1),B點(diǎn)坐標(biāo)(x2,y2),AB斜率是k,則線段AB的垂直平分線斜率是k'
由kk' = -1,所以有
(y1-y2)/(x1-x2) * [(y1+y2)/2 - 0 ]/[(x1+x2)/2 - 6] = -1
(y1² - y2²) / [x1² - x2² -12(x1 - x2)] = -1
代入y1²=2px1,y2²=2px2,化簡(jiǎn)得
2p/(x1 + x2 - 12) = -1
x1 + x2 = 12 - 2p ...(1)
AF²=(x1 - p/2)² + y1² = (x1 - p/2)² + 2px1 = (x1 + p/2)²
AF = x1 + p/2
同理有
BF = x2 + p/2
AF + BF = x1 + x2 + p ...(2)
由(1)、(2)聯(lián)立得12 - 2p + p = 8
即p=4
所以拋物線方程為y² = 8x