已知圆C与y轴相切,圆心c在直线L1:x-5y=0(x> 0)上,且截直线L2:x-y=0的弦长为2倍根号17 (1小题)求圆c的方程 (2小题)若点M(x,y)在圆C上,求x+y的最大值
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![已知圆C与y轴相切,圆心c在直线L1:x-5y=0(x> 0)上,且截直线L2:x-y=0的弦长为2倍根号17 (1小题)求圆c的方程 (2小题)若点M(x,y)在圆C上,求x+y的最大值](/uploads/image/z/13414160-56-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%9C%86C%E4%B8%8Ey%E8%BD%B4%E7%9B%B8%E5%88%87%2C%E5%9C%86%E5%BF%83c%E5%9C%A8%E7%9B%B4%E7%BA%BFL1%EF%BC%9Ax%EF%BC%8D5y%EF%BC%9D0%EF%BC%88x%3E+0%EF%BC%89%E4%B8%8A%2C%E4%B8%94%E6%88%AA%E7%9B%B4%E7%BA%BFL2%EF%BC%9Ax%EF%BC%8Dy%EF%BC%9D0%E7%9A%84%E5%BC%A6%E9%95%BF%E4%B8%BA2%E5%80%8D%E6%A0%B9%E5%8F%B717+%EF%BC%881%E5%B0%8F%E9%A2%98%EF%BC%89%E6%B1%82%E5%9C%86c%E7%9A%84%E6%96%B9%E7%A8%8B+%EF%BC%882%E5%B0%8F%E9%A2%98%EF%BC%89%E8%8B%A5%E7%82%B9M%EF%BC%88x%2Cy%EF%BC%89%E5%9C%A8%E5%9C%86C%E4%B8%8A%2C%E6%B1%82x%EF%BC%8By%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC)
已知圆C与y轴相切,圆心c在直线L1:x-5y=0(x> 0)上,且截直线L2:x-y=0的弦长为2倍根号17 (1小题)求圆c的方程 (2小题)若点M(x,y)在圆C上,求x+y的最大值
已知圆C与y轴相切,圆心c在直线L1:x-5y=0(x> 0)上,且截直线L2:x-y=0的弦长为2倍根号17 (1小题)求圆c的方程 (2小题)若点M(x,y)在圆C上,求x+y的最大值
已知圆C与y轴相切,圆心c在直线L1:x-5y=0(x> 0)上,且截直线L2:x-y=0的弦长为2倍根号17 (1小题)求圆c的方程 (2小题)若点M(x,y)在圆C上,求x+y的最大值
1,因为圆心在直线x-5y=0(x>0)上,可设圆心为(a,a/5)(a>0).
又因为圆与y轴相切,所以半径为a.
由圆截直线x-y=0的弦长为2√17可得:[(a-a/5)/√2]^2+17=a^2,解得:a=5.
圆C的方程为(x-5)^2+(y-1)^2=25.
2,设x+y=z,则y=-x+z.
若圆为(x-5)^2+(y-1)^2=25,则2x^2-(8+2z)x+z^2-2z+1=0.
判别式=(8+2z)^2-8(z^2-2z+1)=0,解得:z=6-5√2或z=6+5√2,x+y的最大值是6+5√2.
1
设圆心O‘(x0,,x0/5),半径R
(x-x0)^2+(y-x0/5)^2=R^2
O'到直线x-y=0距离d
d=|x0-x0/5|/√2=(2√2/5)|x0|
R^2-d^2=(2√17)^2/4=17
R^2=17+(8/25)x0^2
(x-x0)^2+(y-x0/5)^2=17+(8/25)x0^2
弦AB
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1
设圆心O‘(x0,,x0/5),半径R
(x-x0)^2+(y-x0/5)^2=R^2
O'到直线x-y=0距离d
d=|x0-x0/5|/√2=(2√2/5)|x0|
R^2-d^2=(2√17)^2/4=17
R^2=17+(8/25)x0^2
(x-x0)^2+(y-x0/5)^2=17+(8/25)x0^2
弦AB
(x-x0)^2+(x-x0/5)^2=17+(8/25)x0^2
2x^2-(2x0+2x0/5)x+18x0^2/25-17=0
Ax+Bx=x0+x0/5=6x0/5
AxBx=9x0^2/25-17/2
(Ax-Bx)^2=(36/25)x0^2+36x0^2/25-34=(72/25)x0^2-34
Ay-By=Ax-Bx
|AB|^2=2*[(72/25)x0^2-34]=17*4
(72/25)x0^2-34=34
(18/25)x0^2=17
x0^2=17*25/18 (8/25)x0^2=17*4/9
x0=5√(17/18) 或x0=-5√(17/18)
圆方程(x-5√(17/18))^2+(y-√(17/18))^2=17+68/9 或 (x+5√(17/18))^2+(y+√(17/18))^2=17+68/9
2
设x+y=k
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