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Let, $S(\alpha, \beta)$ be the $\color{blue}{\text{focus}}$ and $AB:= lx +my +n=0$ be the $\color{red}{\text{directrix}}$ of a conic. If $P(x,y)$ is a $\color{darkgreen}{\text{generic point}}$ on the conic and $PM \bot AB$ then from the definition of conic, $\color{darkblue}{\text{eccentricity}}$, $e=\frac{SP}{PM}$. It follows that from $PM=\frac{(lx+my+n)}{\sqrt{l^2+m^2}}$ we have: $$SP=e \cdot PM \implies (x-\alpha)^2+(y-\beta)^2=e^2 \frac{(lx+my+n)^2}{l^2+m^2}\qquad(i)$$ It is said be the the general equation of a conic in my book. Although, the above equation is said to be rewritten as $ax^2+2hxy+by^2+2gx+2fy+c=0$ (general second degree equation in x and y), I don't why how it came into play.

To my knowledge, circle is a conic too. It is said in my book, as $\color{darkblue}{\text{eccentricity}}$ of a circle is $0$ if we set $e=0$ then from the above equation $(i)$ the ellipse will become a circle. But I can't see how. Putting the equation in Desmos, I get that it represents a point circle when $e=0$, as expected. Then how does equation $(i)$ which is said be the general equation of conic represents a circle other than a point circle?

I can see pretty easily how $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a circle specifically by setting $a=b$ and $h=0$. But if equation $(i)$ and $ax^2+2hxy+by^2+2gx+2fy+c=0$ are equivalent, why can't equation $(i)$ represent a circle other than point circle?

A related question on this site.

EDIT: Considering a circle is a special case of an ellipse, I have the following equation for a origin centered ellipse where $2a$ is the length of sami-major axis- $$SP= e \cdot PM \implies SP^2= e^2 \cdot PM^2 \implies (x+ae)^2 + y^2 = e^2 \left( \frac{a}{e} +x \right)^2 \\ \implies (x+ae)^2 + y^2 = \frac{e^2}{e^2} a^2 + 2 \frac{a}{e}\cdot x \cdot e^2 + e^2x^2 $$ Now if we want to cancel $e$ in the numerator and the denominator, wouldn't that imply that $e \neq 0$?

Nazmul Hasan Shipon
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  • You are almost entirely correct, except that you forgot to account for the fact that the directrix for a circle must be located at infinity, and hence the distance of any point on the circle to the directrix is infinity. Thus the RHS happens to be in the form $0 \times \infty$ which can give any finite number(which will be the square of the radius). e is not an independent parameter, but a property of the curve, and hence when you change e you must also note the changes you will have to make in its definiton $e = \frac{SP}{PM}$ – First Year in India Oct 21 '21 at 15:02
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    @G12inIndia, the directrix of a circle is located at infinity (not approaching but located). Then the distance as you said is $\infty$. Then shouldn't it be $SP=0 \cdot \infty$ which is undefined? – Nazmul Hasan Shipon Oct 21 '21 at 15:05
  • The concept of an indeterminate form is rather confusing, and I suggest you study it in a bit more detail. However the explanation that I like to use is that no number can ever be equal to infinity, it can only ever tend to infinity and hence you can actually get a finite answer out of that form. To ponder more on such things, I encourage you to think about some other indeterminate forms like $\frac{0}{0}$ and $\frac{1}{0}$ and figure out what they really mean – First Year in India Oct 21 '21 at 15:09
  • @G12inIndia, even if you think in that way, as $e=0$, $0$ times any number should give $0$. – Nazmul Hasan Shipon Oct 21 '21 at 15:12
  • Actually, nope. Like I said the concept will seem a bit unintuitive at first, and I'm not the best person to teach you that. I do recommend Eddie Woo's YouTube channel for this. Try to watch the ones in the Intro to Limits series and maybe also Dividing by Zero. – First Year in India Oct 21 '21 at 15:38
  • See here: https://en.wikipedia.org/wiki/Conic_section#Eccentricity,_focus_and_directrix – Intelligenti pauca Oct 21 '21 at 22:15

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