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设圆G的方程为x~2+y~2=γ~2,则经过圆上一点M(x_0,y_0)的切线的方程是x_0x+y_0y=γ~2,从这条切线的唯一性出发,可得上述命题的三个逆命题:(1)若点M(x_0,y_0)在圆G上,则直线l与圆G相切;(2)若直线l与圆G相切,则点M是切点;(3)若圆心在原点的圆与直线l切于M,则圆为圆G.例1 (课本《解析几何P69第12题)判断直线3x+4y=50与圆x~2+y~2=100
Let the equation of the circle G be x~2+y~2=γ~2, then the equation through the tangent of the point M(x_0,y_0) on the circle is x_0x+y_0y=γ~2, starting from the uniqueness of this tangent line. Three converse propositions of the above propositions are available: (1) If the point M (x_0, y_0) is on the circle G, the line l is tangent to the circle G; (2) If the line l is tangent to the circle G, then the point M is the cut point; (3) If the circle whose center is at the origin and the straight line l are cut at M, then the circle is the circle G. Example 1 (Textbook "Resolving geometry P69 item 12) Judging straight line 3x+4y=50 and circle x~ 2+y~2=100