已知抛物线E:y^=2px（p>0）的准线与x轴交于点M过点M作圆C（x-2）^2+y^2=1的两条切线,切点为A,B,/AB/=4√2/3求抛物线E的方程

Question

已知抛物线E:y^=2px（p>0）的准线与x轴交于点M
过点M作圆C（x-2）^2+y^2=1的两条切线,切点为A,B,/AB/=4√2/3 求抛物线E的方程

Answer 1

M(-p/2,0),C(2,0)

|AB| = 4√2/3
设A在的第一象限,则A的纵坐标 = (4√2/3)/2 = 2√2/3
带入圆的方程,(x - 2)² + (2√2/3)² = 1
(x - 2)² = 1/9
x = 5/3  (舍去x = 7/3 > 2)

圆的半径为r = 1,r² + MA² = MC²
1 + (5/3 + p/2)² + (2√2/3 - 0)² = (2 + p/2)²

p = 2
y² = 4x