圆的切线方程公式证明.doc

解:圆心C(a,b)直线CP的斜率:k1=(y0-b)/(x0-a)因为直线CP与切线垂直,所以切线的斜率:k2=-1/k1=-(x0-a)/(y0-b)根据点斜式,求得切线方程:y-y0=k2(x-x0)y-y0=[-(x0-a)/(y0-b)](x-x0)整理得:(x-x0)(x0-a)+(y-y0)(y0-b)=0(注意:这式也是很好用的切线方程公式)展开后:x0x-ax+ax0+y0y-by+by0-x0²-y0²=0~(1)因为点P在圆上,所以它的坐标满足方程:(x0-a)²+(y0-b)²=r²化简:x0²-2ax0+a²+y1²-2by0+b²=r²移项:-x0²-y0²=-2ax0-2by0+a²+b²-r²~(2)由(2)代入(1),得:x0x-ax+ax0+y0y-by+by0+(-2ax0-2by0+a²+b²-r²)=0化简,(x0x-ax-ax0+a²)+(y0y-yb-by0+b²)=r²整理,(x0-a)(x-a)+(y0-b)(y-b)=r²类似地,对於圆的一般方程:x²+y²+Dx+Ey+F=0,:圆的方程为:x²+y²+Dx+Ey+F=0,圆上一点P(x0,y0)解:圆心C(-D/2,-E/2)直线CP的斜率:k1=(y0+E/2)/(x0+D/2)因为直线CP与切线垂直,所以切线的斜率:k2=-1/k1=-(x0+D/2)/(y0+E/2)根据点斜式,求得切线方程:y-y0=k2(x-x0)y-y0=[-(x0+D/2)/(y0+E/2)](x-x0)整理得:x0x+y0y+Dx/2+Ey/2-Dx0/2-Ey0/2-x0²-y0²=0~(3)因为点P在圆上,所以它的坐标满足方程:x0²+y0²+Dx0+Ey0+F=0移项:-x0²-y0²=Dx0+Ey0+F~(4)由(4)代入(3),得:x0x+y0y+Dx/2+Ey/2-Dx0/2-Ey0/2+Dx0+Ey0+F=0整理,x0x+y0y+D(x+x0)/2+E(y+y0)/2+F=:圆的方程为:(x-a)²+(y-b)²=r²,圆外一点P(x0,y0)解:圆心C(a,b),设切点为M则切线长PM=√(CP²-MC²)(根据勾股定理)=√[(x0-a)²+(y0-b)²-r²](CP:两点间距离公式求得,MC:半径长)类似地,对於圆的一般方程:x²+y²+Dx+Ey+F=0,过圆外的点的切线长....:圆的方程为:x²+y²+Dx+Ey+F=0,圆外一点P(x0,y0)解:圆心C(-D/2,-E/2),设切点为M则切线长PM=√(CP²-MC²)(根据勾股定理)=√[(x0+D/2)²+(y0+E/2)²-((√(D²+E²-4F))/2)²](半径:r=(√(D²+E²-4F))/2)=√(x0²+y0²+Dx0+Ey0+F)酌疗凄蘸庞挠忿倪货锈研司冒皿斥毒镰砖择邯惰太驮尖檬束诧聪双郧