Foundations — Difference of focal radii = 2a property

This page assumes nothing. If the parent note (Difference of focal radii = 2a) used a symbol, we build it here from a picture first. Read top to bottom; each brick sits on the one below it — and no brick uses a symbol built later.

Throughout this page, let $P(x,y)$ be a generic point sitting on the hyperbola — a single dot that is free to slide along the curve. Every symbol we build is a tool for talking about this $P$.

0. A point and its coordinates $(x,y)$

• Picture: look at the red dot. Walk $x$ steps right, then $y$ steps up — you land exactly on it.
• Why the topic needs it: every point $P$ on the hyperbola is written $P(x,y)$. Without an address we cannot measure distances or write an equation.

The special crossing point $(0,0)$ is the origin. The two number-lines are the axes. Later, when both foci sit mirror-image across the origin, this same point is also called the centre of the hyperbola — the balancing point of the whole picture.

1. Distance between two points — the square-root formula

Why a square root appears. Take two points $A(x_1,y_1)$ and $B(x_2,y_2)$. Go from $A$ to $B$ by first moving purely sideways, then purely up. Define those two moves as $\Delta x=x_2-x_1(\text{sideways gap}),\Delta y=y_2-y_1(\text{up gap}).$ These two moves are the legs of a right-angled triangle, and the straight path from $A$ to $B$ is its slanted side (the hypotenuse).

Pythagoras — the oldest rule about right triangles — says $(\text{slanted side}{)}^2=(\text{sideways leg}{)}^2+(\text{up leg}{)}^2.$

• Why the topic needs it: a focal radius (built in §5) is literally the distance from $P$ to a focus.

2. Absolute value $|\cdot |$ — "size, forget the sign"

• Picture: two arrows of equal length pointing opposite ways from $0$ both have the same $|\cdot |$.
• Why the topic needs it: the defining relation is $|SP-S^{\prime }P|=\text{(a fixed gap)}$. The bars matter because on one branch $SP-S^{\prime }P$ is $+$ that gap and on the other it is $-$ that gap; the bars keep the answer a clean positive number for both.

3. The two foci $S$ and $S^{\prime }$, and the centre

• Picture: two red dots, mirror images across the origin (the centre), one on each side.
• Why the topic needs it: the whole property is about distances from these two fixed points. Every measurement in the proof starts at $S$ or $S^{\prime }$. (Their exact $x$-coordinate gets a name in §6, once we have the number that fixes it.)

4. The defining game — a fixed gap builds the curve

Before naming lengths, let us watch the curve being born, so the equation is earned rather than dropped on us.

• Picture: as $P$ slides, both strings to the foci change length, but their difference stays locked at the gap; the trail $P$ leaves is the two-branch curve.
• Why the equation looks the way it does: if you write "distance to $S^{\prime }$ minus distance to $S$ equals a fixed gap," square twice to clear the two roots, and tidy up, the algebra forces the tidy form $\frac{x^2}{(\text{half the gap}{)}^2}-\frac{y^2}{(\text{a second length}{)}^2}=1.$ So the minus sign in the equation is the algebra's memory of the word difference in the rule, and the two denominators are just two squared lengths. The next three sections give those two lengths their proper names.

5. The segment $SP$ — a focal radius

• Picture: two straight strings tied from $P$, one to each red dot. As $P$ slides along the curve, both strings stretch and shrink — but their difference never changes.
• Why the topic needs it: the constant gap is precisely $|r_2-r_1|$.

6. The number $a$, the vertices, and where the foci sit

Now we name the two lengths promised in §4.

• Picture: the two innermost points of the two branches; the distance across them is $2a$.
• Why the topic needs it: the constant difference of focal radii equals this vertex-gap $2a$. That is the headline of the entire topic.

7. Eccentricity $e$ — how "stretched" the curve is, and the foci's address

• Why the topic needs it: the neat result $r_1=|ex-a|$, $r_2=|ex+a|$ is written using $e$. Without $e$ we could not compress the square roots into those tidy forms. See Eccentricity of Conics.

8. The number $b$ and the relation $b^2=a^2(e^2-1)$

$b$ is the second squared length promised in §4 — the number under the $y^2$ term.

• Why the sign is $(e^2-1)$ and not $(1-e^2)$: since $e>1$, the quantity $e^2-1$ is positive, so $b^2$ stays positive (a squared length must be). For an ellipse $e<1$ and the roles flip to $1-e^2$. See b² = a²(e²−1) relation.
• Why the topic needs it: in the proof, substituting this relation is the magic step that makes the messy square roots collapse into perfect squares $(ex\pm a{)}^2$.

9. Putting the symbols together — how the proof reads now

Every symbol you now own lines up exactly with the parent's Step 1. Writing the focal radius $r_1$ with the distance formula (§1) and the foci's address $ae$ (§7), the whole sum sits under one root: $\underset{\text{focal radius (§5)}}{\underbrace{r_1}}=\underset{\text{one distance (§1): entire sum under the bar}}{\underbrace{\sqrt{\underset{\text{coords (§0), }a\text{ (§6), }e\text{ (§7)}}{\underbrace{(x-ae{)}^2}}+y^2}}}.$

The absolute value (§2) shows up in the final $|SP-S^{\prime }P|=2a$ because the two branches swap which focus is farther. Nothing in the parent uses a symbol we have not just built.

Prerequisite map

Read the map in two layers: the top row are low-level building blocks (raw symbols and tools), and everything funnels downward into the single high-level goal TOP.

Equipment checklist

Test yourself — reveal only after answering.

Connections

• Difference of focal radii = 2a property (index 3.4.8)
• Hyperbola — Standard Equation and Definition
• Ellipse — Sum of Focal Radii = 2a
• Eccentricity of Conics
• Focus–Directrix Property
• b² = a²(e²−1) relation