命题1：已知圆O：

http://5b0988e595225.cdn.sohucs.com/images/20181218/00201710bd2d43769291c1c3bd941158.jpeg

上一点M（

http://5b0988e595225.cdn.sohucs.com/images/20181218/a9484fc362a54d0e805817ffcc792004.jpeg

），则以点M为切点的圆的切线方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/2f7262106317443dbe51e86465064da2.jpeg

　　。

　　命题2：已知两相交圆

http://5b0988e595225.cdn.sohucs.com/images/20181218/ba40e652e6ec4fb581e7faa771fa2f81.jpeg

：

http://5b0988e595225.cdn.sohucs.com/images/20181218/78fb93c326754640b6dbe34a47a0f640.jpeg

，圆

http://5b0988e595225.cdn.sohucs.com/images/20181218/319a8d4aab5d4cbeae50a409ecf34f09.jpeg

，则两圆的公共弦所在的直线方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/01dbca2a3f5944ec99cb9d7e2628cab5.jpeg

例、已知点P（

　　）为圆O：

http://5b0988e595225.cdn.sohucs.com/images/20181218/00201710bd2d43769291c1c3bd941158.jpeg

外一点，过点P作圆的切线

http://5b0988e595225.cdn.sohucs.com/images/20181218/5e777bf125734efc9ecfc68870cf0f9a.jpeg

，其中

http://5b0988e595225.cdn.sohucs.com/images/20181218/0eb0203008804e6088a7e5aa1278d149.jpeg

为切点，求切点弦

http://5b0988e595225.cdn.sohucs.com/images/20181218/91dc06675bdb4a369d89b29f735d05e0.jpeg

　　所在的直线方程。

http://5b0988e595225.cdn.sohucs.com/images/20181218/5b24776cc22f42d3bc461e8921d139d5.jpeg

解法1：由题意知

http://5b0988e595225.cdn.sohucs.com/images/20181218/1d4146e3e98c46dca3163be84f15131e.jpeg

所以，O、

http://5b0988e595225.cdn.sohucs.com/images/20181218/ff68bfb2e3e54c4783c7d5a0980ef8f4.jpeg

、P、

http://5b0988e595225.cdn.sohucs.com/images/20181218/e7687878d0e04d0a81f5e60c67f53d69.jpeg

四点共圆

http://5b0988e595225.cdn.sohucs.com/images/20181218/ec93adb6c8bf4d4299f18be491e4099c.jpeg

，且OP为此圆的直径，即圆

http://5b0988e595225.cdn.sohucs.com/images/20181218/ec93adb6c8bf4d4299f18be491e4099c.jpeg

：

http://5b0988e595225.cdn.sohucs.com/images/20181218/a56c963a6a8448f6b21e2a5b83e4493d.jpeg

即

http://5b0988e595225.cdn.sohucs.com/images/20181218/7774d2813bff4f5eaa9291e468b0cde2.jpeg

又

http://5b0988e595225.cdn.sohucs.com/images/20181218/91dc06675bdb4a369d89b29f735d05e0.jpeg

为圆O、圆

http://5b0988e595225.cdn.sohucs.com/images/20181218/ec93adb6c8bf4d4299f18be491e4099c.jpeg

的公共弦，由命题2知，切点弦

http://5b0988e595225.cdn.sohucs.com/images/20181218/91dc06675bdb4a369d89b29f735d05e0.jpeg

所在直线方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/6269f743c3d840b6bdf4b01075ddd4d6.jpeg

　　。

　　解法2：设

http://5b0988e595225.cdn.sohucs.com/images/20181218/2298e9d62307406388ccea286e7ff63d.jpeg

　　由命题1得，

http://5b0988e595225.cdn.sohucs.com/images/20181218/2ddf1948b8a74ef8a15ad7ddea99ebe6.jpeg

方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/ec2b600bdcfb441eb2bbffe73cba717e.jpeg

方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/dae7b56b2d644be68bddd45fe5de0490.jpeg

　　。

　　由

http://5b0988e595225.cdn.sohucs.com/images/20181218/c56b6c0441794d54ad1f615e525d67b3.jpeg

，可得

http://5b0988e595225.cdn.sohucs.com/images/20181218/8f43e6f8c5ff4f5789c5bac31cf59a31.jpeg

http://5b0988e595225.cdn.sohucs.com/images/20181218/c12fc60129594e3b96ec99da23140dd4.jpeg

两点坐标都满足关于

http://5b0988e595225.cdn.sohucs.com/images/20181218/0ca1a62649c143ddb09e5bde53004020.jpeg

的二元一次方程

http://5b0988e595225.cdn.sohucs.com/images/20181218/6269f743c3d840b6bdf4b01075ddd4d6.jpeg

，而过

http://5b0988e595225.cdn.sohucs.com/images/20181218/0eb0203008804e6088a7e5aa1278d149.jpeg

两点的直线有且只有一条，因此，切点弦

http://5b0988e595225.cdn.sohucs.com/images/20181218/91dc06675bdb4a369d89b29f735d05e0.jpeg

所在直线方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/6269f743c3d840b6bdf4b01075ddd4d6.jpeg

　　。

　　解法3：如上图，设

http://5b0988e595225.cdn.sohucs.com/images/20181218/86b2f2bd858e451db179e7b5a3eaf820.jpeg

容易证明

http://5b0988e595225.cdn.sohucs.com/images/20181218/790cd3e8625347888817e2aa4742f9ae.jpeg

，从而M为

http://5b0988e595225.cdn.sohucs.com/images/20181218/91dc06675bdb4a369d89b29f735d05e0.jpeg

　　的中点。

http://5b0988e595225.cdn.sohucs.com/images/20181218/39821e000ba0475cb26276fdb9e1e2fd.jpeg

，M坐标为

　　直线

http://5b0988e595225.cdn.sohucs.com/images/20181218/91dc06675bdb4a369d89b29f735d05e0.jpeg

的方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/a1c55d3be3634dcaaab629160f4a3500.jpeg

　　。

　　即

http://5b0988e595225.cdn.sohucs.com/images/20181218/42b6b3cc6f8249578cfe7a3ce68363b6.jpeg

（*）

　　又由命题1得，

http://5b0988e595225.cdn.sohucs.com/images/20181218/2ddf1948b8a74ef8a15ad7ddea99ebe6.jpeg

方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/6bb1a06f9efc4d00bf119fc58d6db0d7.jpeg

方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/dae7b56b2d644be68bddd45fe5de0490.jpeg

　　。

　　由

http://5b0988e595225.cdn.sohucs.com/images/20181218/c56b6c0441794d54ad1f615e525d67b3.jpeg

，可得

http://5b0988e595225.cdn.sohucs.com/images/20181218/8f43e6f8c5ff4f5789c5bac31cf59a31.jpeg

http://5b0988e595225.cdn.sohucs.com/images/20181218/0b619122bfee457bab2d819b39b3e671.jpeg

代入（*）式得，切点弦

http://5b0988e595225.cdn.sohucs.com/images/20181218/91dc06675bdb4a369d89b29f735d05e0.jpeg

所在直线方程为

http://5b0988e595225.cdn.sohucs.com/images/20181218/6269f743c3d840b6bdf4b01075ddd4d6.jpeg

　　。