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# Find the maximum value of y/x over all real numbers x and y that satisfy

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Find the maximum value of y/x over all real numbers x and y that satisfy $$$(x - 3)^2 + (y - 3)^2 = 6.$$$

waffles  Nov 10, 2017

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(x + 3)2  +  (y - 3)2  =  6          Let's solve this for  y .

(y - 3)2   =   6 - (x + 3)2

y - 3   =   ±√[ 6 - (x + 3)2 ]

y   =   ±√[ 6 - (x + 3)2 ]  +  3           So....using this value for  y....

$$\frac{y}{x}\,=\,\frac{\pm\sqrt{6-(x+3)^2}+3}{x}$$

We can say

$$Y\,=\,\frac{\pm\sqrt{6-(x+3)^2}+3}{x}$$      and we want to know the maximum Y value here.

We can get an approximation by looking at a graph.  (about  5.828)

If you take the derivative and set it = 0, you will get  x = √2 - 2

Then plug this in for  x  and we can find that the exact maximum value  =  3 + 2√2

hectictar  Nov 11, 2017
Sort:

#1
+5254
+1

(x + 3)2  +  (y - 3)2  =  6          Let's solve this for  y .

(y - 3)2   =   6 - (x + 3)2

y - 3   =   ±√[ 6 - (x + 3)2 ]

y   =   ±√[ 6 - (x + 3)2 ]  +  3           So....using this value for  y....

$$\frac{y}{x}\,=\,\frac{\pm\sqrt{6-(x+3)^2}+3}{x}$$

We can say

$$Y\,=\,\frac{\pm\sqrt{6-(x+3)^2}+3}{x}$$      and we want to know the maximum Y value here.

We can get an approximation by looking at a graph.  (about  5.828)

If you take the derivative and set it = 0, you will get  x = √2 - 2

Then plug this in for  x  and we can find that the exact maximum value  =  3 + 2√2

hectictar  Nov 11, 2017

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