已知点a（5根号3.5）.过点a的直线l:x=my+n.若x≤my+n.x-根号3y≥0.y≥0的外接圆的直径为20.求n 急

题目

已知点a（5根号3.5）.过点a的直线l:x=my+n.若x≤my+n.x-根号3y≥0.y≥0的外接圆的直径为20.求n 急

答案

x - √3y = 0与y = 0交于原点O
x = my + n 与y = 0交于点B(n,0)
A(5√3,5)在x = my + n上:5√3 = 5m + n,n = 5(√3 - m)
x = my + n 与x - √3y = 0交于点C(√3n/(√3 - m),n/(√3 - m),即(5√3,5)
OC的中点为D(5√3/2,5/2)
OC的斜率k = 1/√3
OC的中垂线斜率 = -1/k = -√3,中垂线方程:y - 5/2 = -√3(x - 5√3/2) (i)
OB的中点为E(n/2,0)
OB的斜率k' = 0
OB的中垂线方程:x = n/2 (ii)
(i)(ii)联立得圆心E(n/2,10 - √3n/2)
EO² = (20/2)² = 100
(n/2)² + (10 - √3n/2)² = 100
n² - 10√3n = 0
n(n - 10√3) = 0
n = 0,m = √3 (x = my + n 与x - √3y = 0重合,舍去)
n = 10√3