Foundations — Rectangular hyperbola xy = c²

Before you can read the parent note Rectangular hyperbola xy = c² (index 3.4.9), you need to earn every symbol it throws at you. We go from absolute zero. Nothing below assumes you have seen it before.

1. Points and coordinates: $(x,y)$

The fixed centre where you start is called the origin, written $(0,0)$. The two roads you measure along are the axes: the horizontal one is the $x$-axis, the vertical one is the $y$-axis.

Look at the figure. The plum dot sits at $(3,2)$: walk $3$ right, then $2$ up. Every single point on any curve is named this way. The topic needs this because a "curve" is really just the collection of all $(x,y)$ dots that obey one rule.

2. The four quadrants and signs

The two axes chop the sheet into four regions called quadrants, numbered anticlockwise starting from the top-right.

Why the topic needs this: the curve $xy=c^2$ demands $x\times y$ to be a positive number ($c^2$ is a square, always $\ge 0$). Two numbers multiply to a positive result only when they have the same sign. So the curve can only live where $x,y$ agree in sign — Quadrant I (both $+$) and Quadrant III (both $-$). That is exactly why the parent note says the two branches "sit in quadrants I & III". You now know why.

3. The "equals a rule" idea: what an equation of a curve is

When we write $xy=c^2$, the $=$ sign is a filter, not a calculation. It says:

"A point $(x,y)$ is on the curve exactly when its $x$ times its $y$ equals $c^2$. Otherwise it is not."

Everything the parent does ("plug the point in", "check it lies on the tangent") is just this membership test.

4. The multiplication constant $c$ and the square $c^2$

Why squared and not just a letter $k$? Writing the constant as $c^2$ is a deliberate convenience: later the vertices come out as clean $(c,c)$ and the parametric point as $(ct,c/t)$ with no ugly square roots. It is bookkeeping that pays off.

5. The reciprocal / fraction $\frac{c^2}{x}$ and the shape it draws

Solve $xy=c^2$ for $y$: divide both sides by $x$ to get $y=\frac{c^2}{x}$. The bar means divide.

Look at both branches: the burnt-orange branch in Quadrant I and its mirror in Quadrant III. The curve never touches the axes — dividing by $x$ can never reach $y=0$ (you'd need $c^2/x=0$, impossible for finite $x$), and $x$ itself can never be $0$ (division by zero is undefined).

6. Asymptotes: lines the curve approaches but never meets

For $xy=c^2$ the two asymptotes are the axes themselves: the line $x=0$ (the $y$-axis) and $y=0$ (the $x$-axis). In the figure above, as the orange curve races rightward it flattens toward the $x$-axis but stays a whisker above it — that whisker never closes. See Asymptotes of a hyperbola for the general story.

Why the topic needs this: the whole punchline of $xy=c^2$ is that its axes ARE its asymptotes. That is what "referred to its asymptotes" means in the parent's key-facts box.

7. Perpendicular and the word "rectangular"

Two lines are perpendicular when they cross at a right angle ($90^{\circ }$) — like the corner of a book. The $x$- and $y$-axes are perpendicular. Because those axes are also the asymptotes here, the two asymptotes meet at $90^{\circ }$. A hyperbola whose asymptotes are perpendicular is called rectangular (its little corner-boxes are true rectangles). That is the definition the parent opens with.

8. Slope: the steepness number

Why we need slope: a tangent line (a line just grazing the curve at one point) is completely described by two things: where it touches, and how steep it is. The parent finds the tangent's slope to be $-1/t^2$ — a negative number, which the figure shows means "sloping downhill". This matches the mistake-box warning: on each branch, $x$ and $y$ have the same sign, so $-y/x$ is always negative.

9. The parameter $t$: one dial to name every point

Instead of tracking $x$ and $y$ separately, we use a single number $t$ (a "dial") and let it generate both: $x=ct,y=\frac{c}{t}.$

Check it obeys the rule: $x\cdot y=(ct)\cdot \frac{c}{t}=c^2$ ✓ — the $t$'s cancel, so every value of $t$ lands you on the curve. The only forbidden value is $t=0$ (it would make $c/t$ divide by zero).

This covers every case: positive dial, negative dial, and the banned zero.

10. Rotation of axes: tilting your head $45^{\circ }$

The parent claims $xy=c^2$ is the same curve as $x^2-y^2=a^2$, just viewed with tilted axes. Rotating axes means keeping the curve fixed but spinning the two measuring-roads to a new angle. The dots don't move; their coordinate labels change. This is the tool Rotation of axes and it does not distort shape — a circle stays a circle, a hyperbola stays a hyperbola. See also Hyperbola standard form and Eccentricity for where $e=\sqrt{2}$ comes from.

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