圆C:X^2+Y^2-4X-14Y+45=0及点Q(-2,3).(1)P(a,a+1)在圆上,求线段PQ的长及直线PQ的斜率(2)求满足(1)的直线PQ被园C所截得弦AB的长(3)若M为圆C上任一点,求|MQ|的最大值和最小值(4)若实数m,n满足m^2+n^
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![圆C:X^2+Y^2-4X-14Y+45=0及点Q(-2,3).(1)P(a,a+1)在圆上,求线段PQ的长及直线PQ的斜率(2)求满足(1)的直线PQ被园C所截得弦AB的长(3)若M为圆C上任一点,求|MQ|的最大值和最小值(4)若实数m,n满足m^2+n^](/uploads/image/z/8628444-36-4.jpg?t=%E5%9C%86C%3AX%5E2%2BY%5E2-4X-14Y%2B45%3D0%E5%8F%8A%E7%82%B9Q%28-2%2C3%29.%281%29P%28a%2Ca%2B1%29%E5%9C%A8%E5%9C%86%E4%B8%8A%2C%E6%B1%82%E7%BA%BF%E6%AE%B5PQ%E7%9A%84%E9%95%BF%E5%8F%8A%E7%9B%B4%E7%BA%BFPQ%E7%9A%84%E6%96%9C%E7%8E%87%EF%BC%882%EF%BC%89%E6%B1%82%E6%BB%A1%E8%B6%B3%EF%BC%881%EF%BC%89%E7%9A%84%E7%9B%B4%E7%BA%BFPQ%E8%A2%AB%E5%9B%ADC%E6%89%80%E6%88%AA%E5%BE%97%E5%BC%A6AB%E7%9A%84%E9%95%BF%EF%BC%883%EF%BC%89%E8%8B%A5M%E4%B8%BA%E5%9C%86C%E4%B8%8A%E4%BB%BB%E4%B8%80%E7%82%B9%2C%E6%B1%82%7CMQ%7C%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC%EF%BC%884%EF%BC%89%E8%8B%A5%E5%AE%9E%E6%95%B0m%2Cn%E6%BB%A1%E8%B6%B3m%5E2%2Bn%5E)
圆C:X^2+Y^2-4X-14Y+45=0及点Q(-2,3).(1)P(a,a+1)在圆上,求线段PQ的长及直线PQ的斜率(2)求满足(1)的直线PQ被园C所截得弦AB的长(3)若M为圆C上任一点,求|MQ|的最大值和最小值(4)若实数m,n满足m^2+n^
圆C:X^2+Y^2-4X-14Y+45=0及点Q(-2,3).
(1)P(a,a+1)在圆上,求线段PQ的长及直线PQ的斜率
(2)求满足(1)的直线PQ被园C所截得弦AB的长
(3)若M为圆C上任一点,求|MQ|的最大值和最小值
(4)若实数m,n满足m^2+n^2-4m-14n+45=0,求k=(n-3)/(m+2)的最大值和最小值
圆C:X^2+Y^2-4X-14Y+45=0及点Q(-2,3).(1)P(a,a+1)在圆上,求线段PQ的长及直线PQ的斜率(2)求满足(1)的直线PQ被园C所截得弦AB的长(3)若M为圆C上任一点,求|MQ|的最大值和最小值(4)若实数m,n满足m^2+n^
(1)将(a,a+1)带入圆方程,得a=4.
所以点P坐标为(4,5).
PQ斜率为三分之一.PQ=根号[(4+2)^2+(5-3)^2]=2根号10
(3)MQ|的最大值是Q到圆心的距离d再加上圆的半径;|MQ|的最小值是Q到圆心的距离d减去圆的半径.
x²+y²-4x-14y+45=0
(x-2)²+(y-7)²=(2根号2)²
圆心坐标是(2,7),圆的半径是2根号2
Q到圆心的距离是:d=根号[(-2-2)²+(3-7)²]=4根号2
所以|MQ|的最大值是:4根号2+2根号2=6根号2
|MQ|的最小值是:4根号2-2根号2=2根号2
(4)k为直线斜率.
求直线与圆相切时直线的斜率即可,但有两个切线,取较大者;(较小者为最小直.)
x^2+y^2-4x-14y+45=0 ①
y-3=k(x+2) ②
消去y,得
(k^2+1)x^2+4(k^2-2k-1)x+4(k^2-4k+3)=0
有两个相同的根
16(k^2-2k-1)^2-16(k^2+1)(k^2-4k+3)=0
化简得
k^2-4k+1=0
取较大的根
k(max)=2+根号3
k(min)=2-根号3