已知x>0,y>0,且根号X(根号X+根号Y)=3根号Y(根号X+5根号Y)
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已知x>0,y>0,且根号X(根号X+根号Y)=3根号Y(根号X+5根号Y)
求:(1)根号X/根号Y (2)(2x+根号xy+3y)/(x+根号xy-y)
求:(1)根号X/根号Y (2)(2x+根号xy+3y)/(x+根号xy-y)
(1)
√x(√x+√y)=3√y(√x+5√y)
x+√(xy)=3√(xy)+15y
x-2√(xy)-15y=0
(√x+3√y)(√x-5√y)=0
∵x>0 y>0
∴√x>0 √y>0
即:√x+3√y>0
则:√x-5√y=0
√x=5√y
√x/√y=5
(2)
∵√x=5√y
∴x=25y
则2x+√(xy)+3y=2×25y+√(25y²)+3y=50y+5y+3y=58y
而x+√(xy)-y=25y+√(25y²)-y=25y+5y-y=29y
∴[2x+√(xy)+3y]/[x+√(xy)-y]=58y/29y=2
√x(√x+√y)=3√y(√x+5√y)
x+√(xy)=3√(xy)+15y
x-2√(xy)-15y=0
(√x+3√y)(√x-5√y)=0
∵x>0 y>0
∴√x>0 √y>0
即:√x+3√y>0
则:√x-5√y=0
√x=5√y
√x/√y=5
(2)
∵√x=5√y
∴x=25y
则2x+√(xy)+3y=2×25y+√(25y²)+3y=50y+5y+3y=58y
而x+√(xy)-y=25y+√(25y²)-y=25y+5y-y=29y
∴[2x+√(xy)+3y]/[x+√(xy)-y]=58y/29y=2
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