已知点P是抛物线y^2=-4x上一点,设P到此抛物线准线距离是d1,到直线x+y-7=0的距离是已知点P是抛物线y^2=-4x上一点,设P到此抛物线准线距离是d1,到直线x+y-7=0的距离是d2,则d1+d2的最小值是

Question

已知点P是抛物线y^2=-4x上一点,设P到此抛物线准线距离是d1,到直线x+y-7=0的距离是
已知点P是抛物线y^2=-4x上一点,设P到此抛物线准线距离是d1,到直线x+y-7=0的距离是d2,则d1+d2的最小值是

Answer 1

y² = -4x = -2*2x = -2px, p =2
准线 x = 1
P(-p²/4, p)
d1 + d2 = |-p²/4 + p - 7|/√2 + (1 + p²/4)
= |p² - 4p + 28|/(4√2) + 1 + p²/4
= |p² - 4p + 4 + 24|/(4√2) + 1 + p²/4
= (p² - 4p + 28)/(4√2) + 1 + p²/4
= [(√2 + 1)p² - 4p + 28]/(4√2) + 1
p = 4/[2(√2 + 1)] = 2(√2 - 1)时, d1 + d2最小, = 4√2