已知斜率为1的直线l与双曲线c:x^2/a^2- y^2/b^2=1(a〉0,b〉0)相交于B、D两点,且BD的中点为M(1,3)求C的离心率
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![已知斜率为1的直线l与双曲线c:x^2/a^2- y^2/b^2=1(a〉0,b〉0)相交于B、D两点,且BD的中点为M(1,3)求C的离心率](/uploads/image/z/8625373-61-3.jpg?t=%E5%B7%B2%E7%9F%A5%E6%96%9C%E7%8E%87%E4%B8%BA1%E7%9A%84%E7%9B%B4%E7%BA%BFl%E4%B8%8E%E5%8F%8C%E6%9B%B2%E7%BA%BFc%3Ax%5E2%2Fa%5E2%EF%BC%8D+y%5E2%2Fb%5E2%3D1%28a%E3%80%890%2Cb%E3%80%890%29%E7%9B%B8%E4%BA%A4%E4%BA%8EB%E3%80%81D%E4%B8%A4%E7%82%B9%2C%E4%B8%94BD%E7%9A%84%E4%B8%AD%E7%82%B9%E4%B8%BAM%EF%BC%881%2C3%EF%BC%89%E6%B1%82C%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87)
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已知斜率为1的直线l与双曲线c:x^2/a^2- y^2/b^2=1(a〉0,b〉0)相交于B、D两点,且BD的中点为M(1,3)求C的离心率
已知斜率为1的直线l与双曲线c:x^2/a^2- y^2/b^2=1(a〉0,b〉0)相交于B、D两点,且BD的中点为M(1,3)
求C的离心率
已知斜率为1的直线l与双曲线c:x^2/a^2- y^2/b^2=1(a〉0,b〉0)相交于B、D两点,且BD的中点为M(1,3)求C的离心率
斜率为1的直线l:y = x + m
代入双曲线并整理:(b² - a²)x² - 2a²mx -a²(m² + b²) = 0
x₁ + x₂ = 2a²m/(b² - a²)
y₁ + y₂ = x₁ + m + x₂ + m = 2a²m/(b² - a²)+ 2m = 2b²m/(b² - a²)
(x₁ + x₂)/2 = a²m/(b² - a²) = 1
(y₁ + y₂)/2 = b²m/(b² - a²) = 3
相比得 b²/a² = 3
e² = c²/a² = (a² + b²)/a² = 1 + b²/a² = 1 + 3 = 4
e = 2