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In most analytic geometric books the equation of a circle is defined as the special case of a conic $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ when $ a = b \neq 0$ and $h = 0$. Almost all books do not include the condition that the radius should be a real number i.e. $ g^2 + f^2 - ac \geq 0 \Leftrightarrow a\cdot \bigg(\det\begin{vmatrix} a & g & h \\ g & b & f \\ h & f & c \end{vmatrix} \bigg)\leq 0$.

Rather these books consider the case of $ g^2 + f^2 - ac \leq 0 $ as that of an imaginary circle with an imaginary radius. Note that an imaginary circle has no locus on $\mathbb{R}^2$, it in fact represents an empty set.

Most people (that I have met!) argue that the difference is in just opinion, and it makes very little difference to the theory in general. However I have found that the condition a radius should be real adds a lot more checks to the theory: For example if $S_1 = 0$ and $S_2 = 0$ represent the equation to 2 (real) circles, then it is well known that $S_1 + \lambda S_2 = 0$ ($\lambda \neq -1$) represents the equation of a circle which passes through the points of intersection of $S_1 = 0$ and $S_2 = 0$. However, in the proof of the above property it is never checked if the equation of $S_1 + \lambda S_2 = 0$ has real radius! - meaning the equation may not always represent a circle passing through the points of intersection of $S_1 = 0$ and $S_2 = 0$ for all $ \lambda \neq -1$.

As this check does seem important, I would like to know why the concept of an 'imaginary circle' was introduced i.e. any application where the concept of 'imaginary circle' solved/simplified a problem etc.?

Anon
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This approach (common in the 19th century) is still the prevailing point of view in algabraic geometry.

The equation $x^2+y^2 = 1$ is studied as an (algebraic) curve ... which is not the same thing as a point set. Then we may talk about points on the curve ... real points, rational points, complex points, or indeed points where $x,y$ belong to any field of interest.

Similarly, we may study the curve $x^2+y^2 = -1$. In fact it has no real points, but that need not prevent us to talking about complex points or indeed points in some $p$-adic field or finite field.

......

Now I am an analyst, so the point of view above seems alien to me. I prefer to study point sets, and to specify in advance what set (such as $\mathbb R \times \mathbb R$) that I want to take the points from. And I like to talk about curves that are not algebraic, like $y = e^x$ or $\sin x = \sinh y$.

GEdgar
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