Hyperbola — Conjugate and Rectangular

Take the hyperbola \frac{x^2}{9} - \frac{y^2}{4} = 1. Its branches open left and right, hugging two diagonal lines called asymptotes. Now ask: what curve opens up and down, has the exact same asymptotes, and fits into the gaps left behind by the original hyperbola?

That curve is \frac{y^2}{4} - \frac{x^2}{9} = 1. Swap the sign. That is all. The branches that were horizontal become vertical. The asymptotes stay identical. The two hyperbolas together tile the plane around the origin like four wings of a pinwheel, each pair of branches nestled in the gaps left by the other.

This twin is called the conjugate hyperbola. Studying it is not an academic exercise — the relationship between a hyperbola and its conjugate reveals structural properties that you cannot see from one curve alone. And the special case where the two curves are the same shape — the rectangular hyperbola — turns out to be one of the most important curves in coordinate geometry.

The conjugate hyperbola

Start with the standard hyperbola

This curve has its transverse axis along the x-axis, vertices at (\pm a, 0), and branches opening left and right. Its asymptotes are the lines y = \pm \frac{b}{a}x.

The conjugate hyperbola has its transverse axis along the y-axis. Its vertices are at (0, \pm b) and its branches open up and down. The roles of a and b flip: the transverse axis of H' has semi-length b, and the conjugate axis of H' has semi-length a.

Relationship between a hyperbola and its conjugate

The hyperbola and its conjugate are deeply linked. Here is a side-by-side comparison.

Several things are worth noticing.

Same asymptotes. Both H and H' share the exact same pair of asymptotes y = \pm \frac{b}{a}x. This is the defining relationship. The asymptotes are the skeleton that both hyperbolas are built on — one opens between the acute angles of the asymptotes, the other opens between the obtuse angles.

Same value of c. Since c^2 = a^2 + b^2 for both curves, the distance from the centre to each focus is the same number c for both the hyperbola and its conjugate. The foci of H lie on the x-axis; the foci of H' lie on the y-axis; but all four foci sit on a circle of radius c centred at the origin.

Different eccentricities (in general). The eccentricity of H is e = c/a, and the eccentricity of H' is e' = c/b. These are equal only when a = b — a case worth examining closely.

The sum identity. Add the equations of H and H':

The left side is 0 and the right side is 2, so 0 = 2 — which is nonsense. This tells you that no point lies on both H and H'. The two curves never intersect.

The difference identity. Subtract instead:

Those are the asymptotes. So the asymptotes are literally the "average" of the hyperbola and its conjugate — the curve you get when you set the two equations equal.

Eccentricity relation

There is a clean identity linking the eccentricities. Let e = c/a and e' = c/b where c^2 = a^2 + b^2.

So the eccentricities of a hyperbola and its conjugate always satisfy

This is a striking identity. It says that knowing one eccentricity pins down the other. If e = 2, then \frac{1}{4} + \frac{1}{e'^2} = 1, so e' = \frac{2}{\sqrt{3}}.

Rectangular hyperbola: the case a = b

When a = b, the hyperbola \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1 simplifies to

This is called a rectangular hyperbola. The name comes from a geometric fact: its asymptotes are perpendicular.

The asymptotes of a general hyperbola are y = \pm \frac{b}{a}x. When a = b, these become y = x and y = -x. The slopes are +1 and -1. The product of these slopes is (-1), which means the asymptotes are perpendicular. Two perpendicular lines form a rectangle (with the axes), hence the name.

The eccentricity is always \sqrt{2}

For a rectangular hyperbola, a = b, so c^2 = a^2 + b^2 = 2a^2, which gives c = a\sqrt{2}. The eccentricity is

This is a fixed number — it does not depend on the size of the hyperbola. Every rectangular hyperbola, regardless of how large or small, has eccentricity \sqrt{2}. Among all hyperbolas, the rectangular hyperbola is the one closest to being "circular" in spirit (eccentricity as close to 1 as a hyperbola can get while still being a hyperbola, since e > 1 for all hyperbolas).

You can also verify the eccentricity identity: \frac{1}{e^2} + \frac{1}{e'^2} = \frac{1}{2} + \frac{1}{2} = 1. When the hyperbola is rectangular, its conjugate is also rectangular, and both have the same eccentricity \sqrt{2}.

The xy = c^2 form

A rectangular hyperbola x^2 - y^2 = a^2 has perpendicular asymptotes y = x and y = -x. Now imagine rotating the entire figure by 45° so that the asymptotes align with the coordinate axes. What does the equation become?

The rotation

To rotate the coordinate system by 45° anticlockwise, substitute

where (X, Y) are the new coordinates. Substitute into x^2 - y^2 = a^2:

Writing c^2 = \frac{a^2}{2}, this becomes XY = -c^2. If you had instead rotated by 45° clockwise (or equivalently, taken the conjugate hyperbola y^2 - x^2 = a^2), you would get XY = c^2 with a positive sign.

Either way, the equation is of the form xy = k for some constant k. In most textbooks, the standard form is written with the positive constant:

The beauty of this form is its simplicity. The function y = c^2/x is just a scaled version of y = 1/x. Every time you have graphed y = k/x in class, you were graphing a rectangular hyperbola.

Key properties of xy = c^2

Vertices. The vertices are the points closest to the centre (origin). On xy = c^2, the transverse axis is along y = x. Setting y = x gives x^2 = c^2, so x = \pm c. The vertices are (c, c) and (-c, -c).

Eccentricity. Since xy = c^2 is a rectangular hyperbola, its eccentricity is \sqrt{2}, regardless of the value of c.

Foci. The foci lie on the transverse axis y = x, at distance a_{\text{rot}}\sqrt{2} from the centre, where a_{\text{rot}} = c\sqrt{2} is the semi-transverse axis length. So the foci are at (c\sqrt{2}, c\sqrt{2}) and (-c\sqrt{2}, -c\sqrt{2}) — but these calculations are rarely needed in practice.

Parametric form. The curve xy = c^2 has an elegant parametric representation:

For any non-zero t, the point (ct, c/t) lies on xy = c^2 because (ct)(c/t) = c^2. As t varies over all non-zero reals, the point traces the entire curve.

Why xy = c^2 matters

The form xy = c^2 appears naturally in many settings:

• Inverse proportionality. If two quantities x and y are inversely proportional — pressure and volume of a gas (Boyle's law), speed and time for a fixed distance — their graph is xy = k, a rectangular hyperbola.

• Rational functions. The graph of y = \frac{k}{x} is a rectangular hyperbola. Shifts of this, like y = \frac{3}{x - 2} + 1, give rectangular hyperbolas with asymptotes shifted to x = 2 and y = 1.

• Simplicity in computations. Many results about tangents, normals, and chords take their simplest form when the hyperbola is written as xy = c^2, because you avoid the fractions that come with \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.

Worked examples

Common confusions

• "The conjugate hyperbola is just the original rotated by 90°." Only when a = b (the rectangular case). In general, rotating \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 by 90° gives \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1, which is different from the conjugate \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 unless a = b. The conjugate swaps the roles of a and b; rotation does not.

• "A rectangular hyperbola has e = 1." No — e = \sqrt{2} \approx 1.414. The eccentricity of a hyperbola is always greater than 1, and the rectangular case is the smallest it can get among hyperbolas where a = b. The value e = 1 belongs to a parabola.

• "xy = c^2 is a different kind of curve from \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1." They are the same curve — a rectangular hyperbola — just written in different coordinate systems. The form xy = c^2 comes from rotating the axes by 45° so that the asymptotes become the new axes. It is a change of coordinates, not a change of curve.

• "The hyperbola and its conjugate intersect at the vertices of the auxiliary rectangle." They never intersect at all. The auxiliary rectangle has corners at (\pm a, \pm b); checking, \frac{a^2}{a^2} - \frac{b^2}{b^2} = 1 - 1 = 0 \neq 1. Those corners lie on neither curve. The two hyperbolas live in separate regions of the plane.

• "For the conjugate hyperbola, a and b swap." Be careful with the language. The conjugate hyperbola \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 has transverse semi-axis b and conjugate semi-axis a. The values of a and b come from the original equation and do not change — but which one is the transverse axis and which is the conjugate axis does swap.

Going deeper

The tangent-intercept property of xy = c^2

Take any point P = (ct, c/t) on the rectangular hyperbola xy = c^2. The tangent at P can be derived using the parametric form.

Since x = ct and y = c/t:

The tangent at P = (ct, c/t) is:

This line meets the x-axis (set y = 0) at x = 2ct, and the y-axis (set x = 0) at y = 2c/t.

The x-axis intercept is A = (2ct, 0) and the y-axis intercept is B = (0, 2c/t).

Midpoint of AB: \left(\frac{2ct}{2}, \frac{2c/t}{2}\right) = (ct, c/t) = P.

So P is always the midpoint of the intercept segment, regardless of where P sits on the curve. This is a beautiful invariant.

Constant area of the tangent triangle

The triangle OAB — formed by the origin and the two intercepts — has area

The t cancels. The area is 2c^2, a constant independent of P. No matter where you draw the tangent on xy = c^2, the triangle it forms with the axes always has the same area.

This constant-area property is one of the reasons the rectangular hyperbola is a favourite in JEE problems — it gives you a fixed quantity to work with, and many problems reduce to using this fact.

Connection to Bhaskara II's work

The rectangular hyperbola appears implicitly in Bhaskara II's Lilavati (12th century) through problems involving inverse proportion. When Bhaskara writes about quantities where "as one increases, the other decreases in the same ratio," the underlying curve is xy = k — a rectangular hyperbola. The explicit identification of this as a conic section came later, but the curve itself was already present in Indian mathematical thought through practical problems of commerce and astronomy.

Where this leads next

• Hyperbola -- Asymptotes — a full treatment of asymptotes: how to derive them, what they mean geometrically, and why the hyperbola hugs but never touches them.
• Hyperbola -- Parametric and Auxiliary Circle — the parametric form (a\sec\theta, b\tan\theta) for the general hyperbola, the auxiliary circle, the director circle, and the latus rectum.
• Hyperbola -- Introduction — if you need a refresher on the standard form, eccentricity, foci, and basic definitions.
• Ellipse -- Introduction — the sibling conic with e < 1. Many properties of the hyperbola mirror properties of the ellipse, with sign changes.
• Transformation of Axes — the rotation technique used to convert x^2 - y^2 = a^2 into xy = c^2, applied to all conics.