已知抛物线C:y^2=2px(p>0)的焦点为F,其准线为l,P(1⼀2,m)是抛物线C上的一点

2025-02-23 05:53:04
推荐回答(1个)
回答1:

按抛物线的定义,P与准线的距离等于与焦点F(p/2, 0)的距离, PO = PF, 即P为以OF为底的等腰三角形的顶点,P到OF的垂线平方OF,所以OF=P的横坐标的2倍,即p/2 = 1, p = 2

y² = 4x


bx²+9y² = 9b

x²/9 + y²/b = 1

c² = 1 = 9 - b

b = 8

x²/9 + y²/8 = 1


A, B在y轴右侧,

x²/9 + 4x/8 = 1

x = 3/2  (舍去x = -6 < 0)

y = ±√6

渐近线的斜率 = ±√6/(3/2) = ±2√6/3

渐近线: y = ±(2√6/3)x

b/a = 2√6/3, b² = 8a²/3

x²/a² - y²/(8a²/3) = 1

根据对称性,不妨设P在x轴上方, m > 0
y² = 4*1/2 = 2, y = √2

P(1/2, √2)

过P: (1/2)²/a² - (√2)²/(8a²/3) = 1

无解,题似乎有问题