3.4.9 · D2Conic Sections
Visual walkthrough — Rectangular hyperbola xy = c²
We assume you can read a graph and multiply. Everything else — what "asymptote" means, why we rotate, what and are doing — we build here.
Step 1 — Draw the curve we start with
WHAT. Plot every point whose equals the constant .
WHY. This is the rectangular hyperbola — "rectangular" because its two guide-lines (next step) cross at a right angle. We start here because it looks like the hyperbolas you already met in Hyperbola standard form, with the two semi-axes equal ().
PICTURE. Two branches opening left and right, sitting symmetrically about both axes.
In : the term pushes the curve outward horizontally, the pulls it in vertically, and the sum is pinned to the constant . When we get , i.e. — those are the two vertices (red dots), the closest the branches come to the centre.
Step 2 — Find the two lines the curve chases forever
WHAT. For huge and , the "" on the right is tiny compared to and , so the rule is almost . That factors:
WHY. These two lines and are the asymptotes. We hunt for them because — spoiler — we are about to turn our axes to lie exactly on top of them. That is the whole trick. (More on asymptotes in Asymptotes of a hyperbola.)
PICTURE. Two dashed guide-lines crossing at the origin at ; the branches slide along inside them.
The angle between and is exactly — that perpendicular crossing is literally what the word rectangular ("right-angled") is pointing at.
Step 3 — Rotating axes: what "tilt your head " means in symbols
WHAT. Call the new (tilted) coordinates . The Rotation of axes formula rewrites the old in terms of the new :
y = X\sin45^\circ + Y\cos45^\circ = \frac{X+Y}{\sqrt2}.$$ Reading each symbol as it appears: - $X$ = how far along the **new tilted "horizontal"** axis (which lies on $y=x$). - $Y$ = how far along the **new tilted "vertical"** axis (which lies on $y=-x$). - The $\tfrac{1}{\sqrt2}$ everywhere is just $\cos45^\circ$; it shares each new coordinate equally between the old $x$ and old $y$ — that is what a $45^\circ$ turn does. **WHY.** We must express the *old rule* $x^2-y^2=a^2$ using the *new names*, so we need old-in-terms-of-new. That is exactly what these two lines give us. **PICTURE.** The same point, described twice: once by old (blue) axes, once by turned (orange) axes. --- ## Step 4 — Substitute and watch the equation collapse **WHAT.** Put the Step-3 expressions into $x^2 - y^2$: $$x^2 = \left(\frac{X-Y}{\sqrt2}\right)^2 = \frac{(X-Y)^2}{2}, \qquad y^2 = \left(\frac{X+Y}{\sqrt2}\right)^2 = \frac{(X+Y)^2}{2}.$$ Subtract, term by term: $$x^2 - y^2 = \frac{(X-Y)^2 - (X+Y)^2}{2}.$$ Now expand the two squares: - $(X-Y)^2 = X^2 - 2XY + Y^2$ - $(X+Y)^2 = X^2 + 2XY + Y^2$ Subtract — the $X^2$ and $Y^2$ **cancel**, and only the cross-terms survive: $$(X-Y)^2 - (X+Y)^2 = -4XY \;\Rightarrow\; x^2 - y^2 = \frac{-4XY}{2} = -2XY.$$ **WHY.** This cancellation is the *reason the whole trick works*. Rotating by $45^\circ$ is engineered so the "pure square" parts ($X^2$, $Y^2$) vanish and a clean **product** $XY$ is all that's left. **PICTURE.** The cancellation shown as a ledger — squares wipe out, cross-terms remain. So the old rule $x^2 - y^2 = a^2$ becomes, in new axes: $$-2XY = a^2 \quad\Longrightarrow\quad XY = -\frac{a^2}{2}.$$ --- ## Step 5 — The sign, the relabel, and the constant $c^2$ > [!intuition] Where the minus sign goes > $XY = -\tfrac{a^2}{2}$ is negative, so $X$ and $Y$ have **opposite signs** — the branches sit in quadrants II and IV of the *new* frame. That is fine; it is genuinely the same curve. To land on the tidy textbook form with branches in quadrants I and III, we simply flip the orientation of one new axis (rename $Y \to -Y$, allowed — it is still just relabelling addresses). **WHAT.** After that harmless relabel, dropping the capitals back to lowercase for the final tidy names: $$\boxed{\,xy = c^2, \qquad c^2 = \frac{a^2}{2}\,}$$ Reading the result: - $x,y$ = coordinates **along the asymptotes** now (the old asymptotes became the new axes). - $c^2 = \tfrac{a^2}{2}$ = the fixed product every point must obey. It is *half* the old $a^2$ because the rotation shared the size between the two axes. **WHY.** This proves $xy=c^2$ is **not a new curve** — it is $x^2-y^2=a^2$ wearing tilted axes. The asymptotes $y=\pm x$ have literally become the coordinate axes $x=0$ and $y=0$. **PICTURE.** Side by side: left = old curve with dashed asymptotes; right = same curve after we snap the axes onto those asymptotes. > [!formula] What each object turned into > | Before ($x^2-y^2=a^2$) | After ($xy=c^2$) | > |---|---| > | Asymptotes $y=\pm x$ | Axes $x=0,\ y=0$ | > | Vertices $(\pm a,0)$ | Vertices $(c,c),(-c,-c)$ on $y=x$ | > | Constant $a^2$ | Constant $c^2=a^2/2$ | --- ## Step 6 — The degenerate & edge cases (never leave a gap) > [!mistake] Cases you must check before you trust $xy=c^2$ **(a) What if $c=0$?** Then $xy=0$, which means $x=0$ **or** $y=0$ — just the two axes crossing. The hyperbola has collapsed onto its own asymptotes. So $c\neq0$ is required for a genuine curve; equivalently $a\neq0$. This matches $c^2=a^2/2$: zero size in, zero curve out. **(b) Can a point have $x=0$?** No. If $x=0$ then $xy=0\neq c^2$. The curve **never touches the $y$-axis** — that is precisely the asymptote behaviour from Step 2, now visible directly. Same for $y=0$. Every point has *both* coordinates non-zero. **(c) The two branches — both signs covered.** - If $c^2>0$ (real curve) and a point has $x>0$, then $y=c^2/x>0$: **Quadrant I**. - If $x<0$, then $y=c^2/x<0$: **Quadrant III**. - Quadrants II and IV are *empty* (they'd need $xy<0$). This is the graph of the [[Reciprocal function y=1/x|reciprocal function]] scaled by $c^2$. **PICTURE.** The full plane labelled: Q I and Q III carry the curve, Q II and Q IV forbidden, axes forbidden. --- ## Step 7 — Why the name fits: the constant-area rectangle **WHAT.** Take *any* point $P=(x,y)$ on $xy=c^2$. Drop a straight line down to the $x$-axis and across to the $y$-axis. You get a rectangle with the origin. Its width is $|x|$, its height is $|y|$, so: $$\text{area} = |x|\cdot|y| = |xy| = c^2 \quad(\text{constant!}).$$ **WHY.** The equation $xy=c^2$ *is* the statement "the rectangle from here to the axes always has area $c^2$." Move along the curve — the rectangle gets tall-and-thin or short-and-fat, but its **area never changes**. That invariance is the geometric heart of the whole topic. **PICTURE.** Two very different points, two very different rectangles — identical shaded area. --- ## The one-picture summary One figure, the whole journey: start with $x^2-y^2=a^2$ and its perpendicular asymptotes → rotate the ruler $45^\circ$ onto those asymptotes → the squares cancel leaving $xy=-a^2/2$ → relabel to the clean $xy=c^2$ with $c^2=a^2/2$ → and every point makes a constant-area rectangle $c^2$. > [!recall]- Feynman retelling — say it to a friend > We drew a curve, $x^2-y^2=a^2$, that looks like two arches facing away from each other. Far out, the arches run alongside two straight lines that cross at a perfect right angle — that's why we call it "rectangular." Then we did something sneaky: instead of moving the curve, we **turned our graph paper** $45^\circ$ so those two crossing lines *became* the new left–right and up–down axes. When we rewrote the curve's rule in the turned paper, the $x^2$ and $y^2$ pieces cancelled each other out and all that survived was the product $xy$. It came out equal to a fixed number, which we call $c^2$ — and it's exactly half of the old $a^2$ because the turn split the size evenly between the two directions. The payoff picture: from any spot on the curve, the box you make back to the two axes always has the *same area*, $c^2$. Same curve, tilted view, one clean rule: $xy=c^2$. > [!recall]- Quick self-test > Why do the $X^2,Y^2$ terms vanish after rotating $45^\circ$? ::: Because $(X-Y)^2-(X+Y)^2=-4XY$ — the squares subtract away, leaving only the cross term. > Why is $c^2=a^2/2$ and not $a^2$? ::: The $45^\circ$ rotation halves the constant: $-2XY=a^2$ gives $XY=-a^2/2$, so $c^2=a^2/2$. > Why can no point on $xy=c^2$ have $x=0$? ::: Then $xy=0\neq c^2$; the axes are the asymptotes the curve never touches. > What geometric quantity stays constant along the curve? ::: The area $|x||y|=c^2$ of the rectangle from the point to the two axes. --- ## Connections - [[3.4.09 Rectangular hyperbola xy = c² (Hinglish)]] — parent topic (Hinglish). - [[Hyperbola standard form]] — the $a=b$ starting curve $x^2-y^2=a^2$. - [[Rotation of axes]] — the exact tool of Steps 3–4. - [[Asymptotes of a hyperbola]] — the lines that become the new axes. - [[Reciprocal function y=1/x]] — the shape you get once it's $xy=c^2$. - [[Eccentricity]] — why $e=\sqrt2$ for this equilateral curve. - [[Tangent and Normal to conics]] — the next thing you build on $xy=c^2$.