Foundations — Circle as degenerate conic (e = 0)

Before you can believe the parent page, you need every letter it throws at you to already mean something. This page builds them one at a time, each on top of the last, each anchored to a picture.

---

0. What a point and its coordinates even are

Why we need this. Every equation on the parent page — $x^2+y^2=a^2$, $(x-h{)}^2+(y-k{)}^2=r^2$ — is a rule that some points obey and others don't. Without knowing what $(x,y)$ names, those rules are meaningless squiggles. Look at the figure: the blue dot sits at $x=3$ to the right and $y=2$ up, so it is named $P(3,2)$.

The plane is split into four quadrants by the two axes. We will care about all four later, so meet them now: top-right, top-left, bottom-left, bottom-right — signs $(+,+),(-,+),(-,-),(+,-)$.

---

1. Distance between two points — the measuring stick

Everything about a circle is about distance, so we need one honest way to measure it.

Why this exact tool, not another? A ruler only works along a straight axis. The two gaps $\Delta x,\Delta y$ are along the axes, but the line we want is slanted. Pythagoras is the only elementary rule that converts two perpendicular gaps into one slanted length — that is precisely the question we're asking. This is why the parent page says the circle equation is "built with the Distance Formula".

---

2. Squaring, and why we're allowed to square both sides

The symbol $x^2$ means $x\times x$. Two facts we lean on constantly:

• A square is never negative: $x^2\ge 0$ for every real $x$. (A length times itself can't be negative.)
• Because both a distance and a radius are $\ge 0$, the statement "$\sqrt{\text{stuff}}=r$" and the statement "$\text{stuff}=r^2$" are exactly the same — squaring loses no information when both sides are already non-negative.

Why this matters. The parent page writes $\sqrt{(x-h{)}^2+(y-k{)}^2}=r⟹(x-h{)}^2+(y-k{)}^2=r^2,$ and calls squaring "allowed, both sides $\ge 0$." That one-line justification is this rule. Squaring simply clears the ugly root so the equation is a polynomial we can expand.

---

3. Radius $r$ and centre $C$ — what a circle is

• $C(h,k)$: the centre. $h$ = its $x$-coordinate, $k$ = its $y$-coordinate. (Letters $h,k$ are just "the centre's address.")
• $r$: the radius, the same distance in every direction — this sameness in all directions is the whole point.

Why the equation looks the way it does. "Distance from $P(x,y)$ to $C(h,k)$ equals $r$" is, word for word, the distance formula set equal to $r$: $\sqrt{(x-h{)}^2+(y-k{)}^2}=r\stackrel{\text{square}}{\to }(x-h{)}^2+(y-k{)}^2=r^2.$ Notice: no directrix, no special direction appears — because a circle has none. Hold that thought for $e=0$.

---

4. Eccentricity $e$ — the "how squashed" dial

Read $e$ as a squash dial, from the Eccentricity note:

$e$	Shape	Feel
$0$	circle	not squashed at all — round
$0<e<1$	Ellipse	a gentle oval
$1$	Parabola	just barely open
$>1$	Hyperbola	wide-open two-branch

Why this ratio, of all things? One number $e$ classifies the entire conic family from a single common definition. That economy is why the parent page can say "a circle is just the conic with $e=0$" instead of treating the circle as an unrelated special shape.

---

5. Semi-axes $a$, $b$ and the ellipse identity

An ellipse (oval) has a long half-width and a short half-width:

• $a$ = semi-major axis = half of the longest diameter.
• $b$ = semi-minor axis = half of the shortest diameter.
• $c$ = distance from centre to each focus (an ellipse has two foci, at $(\pm c,0)$).

They are tied together by the ellipse identity the parent page uses: $e=\frac{c}{a},b^2=a^2(1-e^2).$

Read it as a machine. Feed in how squashed the ellipse is ($e$) and its long half-width ($a$); out comes its short half-width ($b$). When $e$ is big the bracket $(1-e^2)$ is small, so $b$ is much smaller than $a$ — very squashed. When $e=0$ the bracket is $1$, so $b=a$: both half-widths equal → round → circle. That is the arithmetic heart of "$e=0$ is a circle".

---

6. Limits and $\infty$ — the careful way to reach $e=0$

A limit, written $e\to 0$, means "watch what a quantity settles toward as $e$ slides closer and closer to $0$, without ever demanding $e$ equals $0$ outright." The symbol $\infty$ ("infinity") is not a number; "$\to \infty$" means "grows without any bound".

Why we need a limit here. Substituting $e=0$ broke the shape into a single point (Section 4). Sliding $e\to 0$ instead lets us watch the ellipse relax smoothly into a circle:

• $c=ae\to 0$: the two foci slide together into the centre.
• $b^2=a^2(1-e^2)\to a^2$, so $b\to a$: the axes become equal.
• the directrix sits at $x=a/e$, and as $e\to 0$ this runs off to $\infty$: the circle ends up with no directrix.

This is exactly the parent page's three-step derivation — now every symbol in it is one you own.

---

Prerequisite map

---

Equipment checklist

---

Connections

• 3.4.10 Circle as degenerate conic (e = 0) (Hinglish) — same page, Hinglish.
• Conic Sections — the family these foundations serve.
• Distance Formula — Section 1, our measuring stick.
• Eccentricity — Section 4, the squash dial $e$.
• Ellipse — Section 5, the shape a circle relaxes from.
• Parabola and Hyperbola — the other $e$-values.
• Completing the Square — needed to turn general form into $(x-h{)}^2+(y-k{)}^2$.
• Degenerate Conics — where the point-circle limit lands.