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If equations of two curves in a plane are given as

$$ f(x,y)=0,\, g(x,y)=0,\, $$

what geometric interpretation can be given to the following five derived curves i.e., when seen plotted together in relation to individual $f,g?$

$$ f(x,y - g(x,y)=0,\, \tag1 $$

Particularly if $f,g$ are circles the above represents their radical axis. Next,

$$ f(x,y) + g(x,y)=0,\, \tag2$$

and with parameter $\lambda$

$$ f(x,y)+ \lambda \, g(x,y)=0,\, \tag3$$

$$ f(x,y) \cdot g(x,y)=1,\, \tag4$$

$$\frac{ 2 f(x,y)\cdot g(x,y) } {f(x,y) + g(x,y)}=1. \tag5$$

Narasimham
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  • Did you mean the curves are $y=f(x)$ and $y=g(x)$ or $f(x,y)=0$ and $g(x,y)=0$? – Ng Chung Tak May 05 '17 at 09:17
  • Yes, that's what I mean, thanks much for the suggestion,would be changing it. – Narasimham May 05 '17 at 12:38
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    Your comment about the radical axis is not true in general. For example, there are many ways to construct a function $f$ such that $f(x,y)=0$ is the equation of the unit circle; only one of these many functions is a quadratic in $x$ and $y.$ Did you forget to mention some restriction on what kind of functions $f$ and $g$ can be? – David K May 05 '17 at 13:13
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    I don't think you understood the previous comment. A more concrete example: Let $f(x,y) = x^2 + y^2-1,$ $g(x,y) = ((x-1)^2 + y^2)^2-1.$ Now what does $f(x,y)-g(x,y)=0$ describe? – David K May 05 '17 at 13:48
  • @DavidK a circle with a bite... :D – Rafa Budría May 05 '17 at 13:52
  • @DavidK Whatever may be parametric definition ( e.g., hyperbolic) , a circle in the plane remains the same. My comment before typo correction equn (1) is that $ x^2+y^2 + ax+by+c -(x^2+y^2 +Ax+By+C)= 0 $ represents a radical axis in all 3 cases of intersection. – Narasimham May 05 '17 at 14:21
  • So you are assuming that $f$ and $g$ will always be quadratic functions of $x$ and $y.$ That is one unstated assumption. You are also making some additional assumptions, for example even though $g_1(x,y)=x^2+y^2+3x+4y+20=0$ and $g_2(x,y)=2x^2+2y^2+6x+8y+40=0$ are equations of the same circle, you will always use $g_1$ and never $g_2,$ because while $x^2+y^2-10-g_1(x,y)=0$ is a line, $x^2+y^2-10-g_2(x,y)=0$ is a circle. But in other cases, how do you decide which formula is correct to use? For example, $x^2+\frac12y^2-1=0$ versus $2x^2+y^2-2=0.$ – David K May 05 '17 at 15:22
  • I am not making any assumptions about $f,g$ except they are smooth differentiable and so on. In the second example you need to divide by 2 and subtract to get to radical axis. Else you get another circle sharing same fixed points of intersection. – Narasimham May 05 '17 at 15:40
  • Smooth differentiable is a big assumption. "And so on" could be a lot more assumptions. And if you're willing to multiply the function by whatever constant is needed in order to make the answer come out to what you want, then it is pointless to distinguish $f(x,y)-g(x,y)$ or $f(x,y)+\lambda g(x,y)$ from $f(x,y)+g(x,y).$ – David K May 06 '17 at 00:21

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