please help with analytic geometry if you can
posted by Knights .
Find the maximum value of y/x over all real numbers x and y that satisfy (x3)^2+(y3)^2 = 6.
It is a circle, but how do we even begin? As a matter of fact, how is there not only like 1 solution??

it is a circle centered at (3,3) with radius √6. So, the maximum y = 3+√6.
The minimum x is likewise 3√6.
Unfortunately, the two values don't fit a point on the circle.
If you have calculus, you realize that
y/x = (3+√(6(x3)^2))/x
= (3+√(x^2+6x3))/x
y/x has a maximum where
3(√(x^2+6x3)+x1)/(x^2√(x^2+6x3)) = 0
Since the bottom is never zero, we want
√(x^2+6x3)+x1 = 0
x^2+6x3 = x^22x+1
x = 2±√2
y/x at x = 2√2 is 3+2√2 = 5.82
Not sure how you'd find this result without calculus. It's not clear that the maximum y/x occurs just near the leftmost point of the circle. It's clearly near the minimum x, but it's not obvious just how clear. 
Great Question!
I agree with Steve's answer, although I obtained mine using a slightly different approach
I found dy/dx of the original equation to be
dy/dx = (x3)/(3y)
I then let M = y/x
by the quotient rule,
dM/dx = (x dy/dx  y)/x^2
setting this equal to zero for a max of M
I got dy/dx = y/x (very interesting)
equating the two versions of dy/dx and simplifying gave me
x^2 + y^2 = 3x + 3y
after expanding the original circle equation and replacing x^2 + y^2
I ended up with
x+y = 4 > y = 4x
I finally subbed this back into x^2+y^2 = 3x3y
and got Steve's answer of
x = 2 ± √2
When x = 2+√2 , y = 2√2, and y/x = .5857..
which would be a minimum
when x = 2  √2 , y = 2+√2 , and y/x = 5.8284...
the maximum
Did you notice the symmetry of the x and y values for the max and min ?