已知圆C:x^2+y^2+2x-4y+3=0.从圆外一点P(X1,Y1)向圆引一条切线,切点为M,O为坐标原点,且有┃PM┃=┃PO┃,求使┃PM┃最小的P点坐标.哪个大侠帮忙的?
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/20 11:06:14
![已知圆C:x^2+y^2+2x-4y+3=0.从圆外一点P(X1,Y1)向圆引一条切线,切点为M,O为坐标原点,且有┃PM┃=┃PO┃,求使┃PM┃最小的P点坐标.哪个大侠帮忙的?](/uploads/image/z/9451103-23-3.jpg?t=%E5%B7%B2%E7%9F%A5%E5%9C%86C%EF%BC%9Ax%5E2%2By%5E2%2B2x-4y%2B3%3D0.%E4%BB%8E%E5%9C%86%E5%A4%96%E4%B8%80%E7%82%B9P%EF%BC%88X1%2CY1%29%E5%90%91%E5%9C%86%E5%BC%95%E4%B8%80%E6%9D%A1%E5%88%87%E7%BA%BF%2C%E5%88%87%E7%82%B9%E4%B8%BAM%2CO%E4%B8%BA%E5%9D%90%E6%A0%87%E5%8E%9F%E7%82%B9%2C%E4%B8%94%E6%9C%89%E2%94%83PM%E2%94%83%3D%E2%94%83PO%E2%94%83%2C%E6%B1%82%E4%BD%BF%E2%94%83PM%E2%94%83%E6%9C%80%E5%B0%8F%E7%9A%84P%E7%82%B9%E5%9D%90%E6%A0%87.%E5%93%AA%E4%B8%AA%E5%A4%A7%E4%BE%A0%E5%B8%AE%E5%BF%99%E7%9A%84%3F)
已知圆C:x^2+y^2+2x-4y+3=0.从圆外一点P(X1,Y1)向圆引一条切线,切点为M,O为坐标原点,且有┃PM┃=┃PO┃,求使┃PM┃最小的P点坐标.哪个大侠帮忙的?
已知圆C:x^2+y^2+2x-4y+3=0.从圆外一点P(X1,Y1)向圆引一条切线,切点为M,O为坐标原点,且有┃PM┃=┃PO┃,求使┃PM┃最小的P点坐标.
哪个大侠帮忙的?
已知圆C:x^2+y^2+2x-4y+3=0.从圆外一点P(X1,Y1)向圆引一条切线,切点为M,O为坐标原点,且有┃PM┃=┃PO┃,求使┃PM┃最小的P点坐标.哪个大侠帮忙的?
x²+y²+2x-4y+3=0
(x+1)²+(y-2)²=2
圆心C(-1,2),半径√2(如图)
┃PM┃=┃PO┃,则┃PM┃²=┃PO┃²
┃PM┃²=┃PC┃²-┃CM┃²=┃PO┃²
(X1+1)²+(Y1-2)²-2=X1²+Y1²
2X1-4Y1+3=0,2X1=4Y1-3
求┃PM┃最小值,即┃PO┃=√(X1²+Y1²)最小值
另S=√(X1²+Y1²
4S²=4(X1²+Y1²)=(2X1)²+4Y1²=(4Y1-3)²+4Y1²=20(Y1-3/5)²+9/5
4S² ≥9/5
S ≥3√5/10
此时Y1=3/5,X1=-3/10
所以P点坐标(-3/10,3/5)
┃PM┃最小值为3√5/10