已知双曲线x²/a²-y²/b²=1(a>0,b>0)的两条渐近线均和圆C:x²+y²-6x+5=0相切,且双曲线的右焦点为圆C的圆心,则该双曲线的方程为
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![已知双曲线x²/a²-y²/b²=1(a>0,b>0)的两条渐近线均和圆C:x²+y²-6x+5=0相切,且双曲线的右焦点为圆C的圆心,则该双曲线的方程为](/uploads/image/z/9298677-21-7.jpg?t=%E5%B7%B2%E7%9F%A5%E5%8F%8C%E6%9B%B2%E7%BA%BFx%26%23178%3B%2Fa%26%23178%3B-y%26%23178%3B%2Fb%26%23178%3B%3D1%28a%3E0%2Cb%3E0%29%E7%9A%84%E4%B8%A4%E6%9D%A1%E6%B8%90%E8%BF%91%E7%BA%BF%E5%9D%87%E5%92%8C%E5%9C%86C%3Ax%26%23178%3B%2By%26%23178%3B-6x%2B5%3D0%E7%9B%B8%E5%88%87%2C%E4%B8%94%E5%8F%8C%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%8F%B3%E7%84%A6%E7%82%B9%E4%B8%BA%E5%9C%86C%E7%9A%84%E5%9C%86%E5%BF%83%2C%E5%88%99%E8%AF%A5%E5%8F%8C%E6%9B%B2%E7%BA%BF%E7%9A%84%E6%96%B9%E7%A8%8B%E4%B8%BA)
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已知双曲线x²/a²-y²/b²=1(a>0,b>0)的两条渐近线均和圆C:x²+y²-6x+5=0相切,且双曲线的右焦点为圆C的圆心,则该双曲线的方程为
已知双曲线x²/a²-y²/b²=1(a>0,b>0)的两条渐近线均和圆C:x²+y²-6x+5=0相切,且双曲线的右焦点为圆C的圆心,则该双曲线的方程为
已知双曲线x²/a²-y²/b²=1(a>0,b>0)的两条渐近线均和圆C:x²+y²-6x+5=0相切,且双曲线的右焦点为圆C的圆心,则该双曲线的方程为
因为圆C:x²+y²-6x+5=0⇔(x-3)²+y²=4,由此知道圆心C(3,0),圆的半径为2,
又因为双曲线的右焦点为圆C的圆心而双曲线x²/a²-y²/b²=1(a>0,b>0),
∴a²+b²=9 ①
又双曲线的两条渐近线均和圆C:x²+y²-6x+5=0相切,而双曲线的渐近线方程为:
y=±b/a x ⇒bx±ay=0⇒3b/√(a²+b²)=2 ②
联立①②得:b=2,a²=5.
∴双曲线的方程:x²/5-y²/4=1.