Help Me Solve Equationsystem Problem with x & y

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The discussion revolves around solving a system of equations involving x and y, defined by two equations that represent the upper hemispheres of circles. The equations can be simplified by squaring both sides and rearranging to understand their geometric representation in the xy-plane. Substituting one equation into the other leads to complex results that are challenging to solve algebraically. The suggestion is to approach the problem geometrically for a clearer understanding of potential intersections based on the constants involved. Ultimately, the focus is on finding a solution or confirming the feasibility of the equations given the constants a, b, c, d, e, and f.
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My maths skills are so rusty that I can't figure out how I simplify these equations so that I get a formula for x and y... a,b,c,d,e,f are constants

y=\sqrt{b^{2} - (x-f)^{2}} + e
x=\sqrt{a^{2} - (y-c)^{2}} + d

Can anyone help me? And is this equationsystem even possible?
 
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If you subtract the constants from both sides and square both sides, you should be able to see that your equations can be graphed in the xy-plane as the upper hemisphere of a circle of radius b centered at (f, e) and the upper hemisphere of a circle of radius a centered at (d, c). Whether these two curve segments intersect or not is up to the values of the constants.
To start, you can just use substitution: substitute your expression for y as a function of x into the second equation.
 
I tried substituting y as a function of x into the second equation but I got an awfully complicated equation which I was unable to solve as I'm not that good at maths... :( Are you able to get a solution?
 
nastyjoe said:
My maths skills are so rusty that I can't figure out how I simplify these equations so that I get a formula for x and y... a,b,c,d,e,f are constants

y=\sqrt{b^{2} - (x-f)^{2}} + e
x=\sqrt{a^{2} - (y-c)^{2}} + d

Can anyone help me? And is this equationsystem even possible?

nastyjoe said:
I tried substituting y as a function of x into the second equation but I got an awfully complicated equation which I was unable to solve as I'm not that good at maths... :( Are you able to get a solution?

Welcome to the PF.

What are these equations from?
 
If you square the first equation, you get
##(y-e)^2 + (x-f)^2 = b^2##

If you draw a graph of that equation, what shape of curve do you get? (If you can't see the answer to that, start with the simpler case when e = f = 0).

The easiest way to solve the two equations is using geometry, not algebra.
 

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