过圆x²+y²=r²外一点M(x0,y0)向圆引切线,设切点为A,B,求证:直线AB的方程是xx0+yy0=r².
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![过圆x²+y²=r²外一点M(x0,y0)向圆引切线,设切点为A,B,求证:直线AB的方程是xx0+yy0=r².](/uploads/image/z/5182747-43-7.jpg?t=%E8%BF%87%E5%9C%86x%26%23178%3B%2By%26%23178%3B%3Dr%26%23178%3B%E5%A4%96%E4%B8%80%E7%82%B9M%28x0%2Cy0%29%E5%90%91%E5%9C%86%E5%BC%95%E5%88%87%E7%BA%BF%2C%E8%AE%BE%E5%88%87%E7%82%B9%E4%B8%BAA%2CB%2C%E6%B1%82%E8%AF%81%3A%E7%9B%B4%E7%BA%BFAB%E7%9A%84%E6%96%B9%E7%A8%8B%E6%98%AFxx0%2Byy0%3Dr%26%23178%3B.)
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过圆x²+y²=r²外一点M(x0,y0)向圆引切线,设切点为A,B,求证:直线AB的方程是xx0+yy0=r².
过圆x²+y²=r²外一点M(x0,y0)向圆引切线,设切点为A,B,求证:直线AB的方程是xx0+yy0=r².
过圆x²+y²=r²外一点M(x0,y0)向圆引切线,设切点为A,B,求证:直线AB的方程是xx0+yy0=r².
如图
以M点为圆心,MB为半径做圆
则AB为两个圆的公共弦
根据勾股定理,圆M的半径为sqrt((x0)^2+(y0)^2-r^2)
则M的方程为(x-x0)^2+(y-y0)^2=(x0)^2+(y0)^2-r^2
两个圆的方程相减即得AB的方程
((x-x0)^2+(y-y0)^2)-(x²+y²)=(x0)^2+(y0)^2-2*r^2
化简得
xx0+yy0=r².