求椭圆的焦半径公式推导
来源:学生作业帮 编辑:搜搜做题作业网作业帮 分类:数学作业 时间:2024/07/04 12:35:04
求椭圆的焦半径公式推导
![求椭圆的焦半径公式推导](/uploads/image/z/19744548-60-8.jpg?t=%E6%B1%82%E6%A4%AD%E5%9C%86%E7%9A%84%E7%84%A6%E5%8D%8A%E5%BE%84%E5%85%AC%E5%BC%8F%E6%8E%A8%E5%AF%BC)
证明:
|PF1|²
=(x - c)² + y²
=[a²(x - c)² + a²y²]/a²
=[a²x² - 2a²cx + a²c² + a²y²]/a² /***--根据b²x² + a²y² = a²b² ***/
=[a²x² - 2a²cx + a²c² + a²b² - b²x²]/a²
=[(a²-b²)x² = 2a²cx + a²(b² + c²)]/a²
=[c²x² -2a²cx + a^4]/a²
=(a² - cx)²/a²
∴PF1 = (a² - cx)/a = a - (c/a)x = a - ex
同理可证:PF2 = a + ex
|PF1|²
=(x - c)² + y²
=[a²(x - c)² + a²y²]/a²
=[a²x² - 2a²cx + a²c² + a²y²]/a² /***--根据b²x² + a²y² = a²b² ***/
=[a²x² - 2a²cx + a²c² + a²b² - b²x²]/a²
=[(a²-b²)x² = 2a²cx + a²(b² + c²)]/a²
=[c²x² -2a²cx + a^4]/a²
=(a² - cx)²/a²
∴PF1 = (a² - cx)/a = a - (c/a)x = a - ex
同理可证:PF2 = a + ex