Hyperbola — Advanced

Imagine standing on a hilltop and looking at two straight roads that stretch into the distance. Each road just barely touches a curved lake at one point before veering away. Someone asks: what is the shortest path from one touching point to the other, staying over the lake? That path — the chord connecting the two tangent points — is the chord of contact, and it shows up everywhere in conic section problems.

Stand at the point (6, 3) and look at the hyperbola \dfrac{x^2}{9} - \dfrac{y^2}{4} = 1. You are outside the curve — you can verify: \dfrac{36}{9} - \dfrac{9}{4} = 4 - 2.25 = 1.75 > 1, which means the point lies in the exterior. From where you stand, you can draw exactly two lines that just graze the curve without cutting through it. Those are the two tangent lines from your point.

Now connect the two points where those tangent lines touch the hyperbola. That connecting line is the chord of contact. And there is something remarkable about it: its equation has exactly the same form as the tangent equation, using the external point's coordinates. The pattern that governs tangent lines — the T = 0 substitution — governs the chord of contact too.

This article covers four constructions that extend the tangent theory: chord of contact, chord with a given midpoint, the pair of tangents from an external point, and the reflection property. Each one follows from the same algebraic machinery, and each one appears regularly in JEE and competitive problems. Underlying all of them is a single unifying idea — the pole-polar correspondence — which ties the whole theory together.

The notation

Before the derivations, set up the shorthand that makes the algebra readable. For the hyperbola S \equiv \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} - 1 = 0:

• S_1 = \dfrac{x_1^2}{a^2} - \dfrac{y_1^2}{b^2} - 1 — the value of S at the point (x_1, y_1).
• T = \dfrac{x x_1}{a^2} - \dfrac{y y_1}{b^2} - 1 — the expression you get by replacing x^2 \to x x_1 and y^2 \to y y_1.

If (x_1, y_1) is on the hyperbola, S_1 = 0 and T = 0 is the tangent. If (x_1, y_1) is outside the hyperbola, S_1 > 0 and T = 0 is the chord of contact. If (x_1, y_1) is inside the hyperbola, S_1 < 0.

This notation is shared across all conics — circles, parabolas, ellipses, and hyperbolas all use S, S_1, and T in the same way.

Chord of contact

Suppose you are at an external point (x_1, y_1) — meaning S_1 > 0. You draw the two tangents from this point, touching the hyperbola at points A and B. The line AB is the chord of contact.

Derivation. Let A = (\alpha_1, \beta_1) and B = (\alpha_2, \beta_2) be the two contact points, both on the hyperbola. The tangent at A is:

Since this tangent passes through (x_1, y_1):

Similarly, the tangent at B passes through (x_1, y_1):

Equations (1) and (2) say that both (\alpha_1, \beta_1) and (\alpha_2, \beta_2) satisfy the equation:

where X and Y are the running coordinates. But the line through two points is unique. So the chord of contact AB has the equation:

The elegance here is that the same formula does double duty. When the point is on the curve, the formula gives the tangent. When the point is off the curve, the same formula gives the chord of contact. The algebra does not care — the equation T = 0 adapts to the context.

Chord with a given midpoint

Here is a different question: you are told that a chord of the hyperbola has its midpoint at the point (h, k). What is the equation of that chord?

Derivation. Let the endpoints of the chord be (x_1, y_1) and (x_2, y_2), both on the hyperbola. Then:

Subtract:

The midpoint is (h, k), so x_1 + x_2 = 2h and y_1 + y_2 = 2k. The slope of the chord is m = \dfrac{y_1 - y_2}{x_1 - x_2}. Substituting:

The chord passes through (h, k) with slope \dfrac{b^2 h}{a^2 k}:

Multiply by a^2 k:

Divide by a^2 b^2:

The left side is T (with (h, k) playing the role of (x_1, y_1), minus the -1). The right side is S_1 (plus 1). More precisely:

The formula T = S_1 is universal across conics. For any conic, the chord whose midpoint is a given point has the equation T = S_1, where T and S_1 are computed from the conic's equation.

Pair of tangents from an external point

From an external point (x_1, y_1), two tangent lines go to the hyperbola. The combined equation of both tangent lines together — as a single second-degree equation — has a beautiful form.

Derivation. Consider a general point (x, y) in the plane. Draw the line from (x_1, y_1) through (x, y). A general point on this line can be parametrised as:

where t = 0 gives (x_1, y_1) and t = 1 gives (x, y). Substitute this into the hyperbola equation S = 0:

Expand each squared term. For the x-part:

Similarly for the y-part. Collecting powers of t:

where S_{(x,y)} denotes S evaluated at (x, y) — which we simply call S — and T_{(x,y)} is T evaluated using (x_1, y_1) as the fixed point. More precisely, after careful expansion, the quadratic in t becomes:

This quadratic has two roots in t, corresponding to the two points where the line from (x_1, y_1) through (x, y) meets the hyperbola. For the line to be tangent, these two intersection points must coincide, so the discriminant of this quadratic in t must be zero:

Now, the locus condition simplifies. Since (x_1, y_1) is the external point (a fixed point, not a variable), S_1 is a constant. The condition that a point (x, y) lies on one of the tangent lines from (x_1, y_1) is:

This equation, SS_1 = T^2, is a second-degree equation in x and y. It represents two straight lines (the pair of tangents) taken together. The equation T^2 = SS_1 is a factored form — in principle, you can factor the left minus the right into two linear factors, and those are the individual tangent lines.

The formula works for all conics. For a circle x^2 + y^2 = r^2, for a parabola y^2 = 4ax, for an ellipse — the same SS_1 = T^2.

The reflection property

This is the property that makes hyperbolas useful in physics and engineering.

Statement. If P is any point on the hyperbola and S, S' are the two foci, then the tangent at P bisects the exterior angle between the focal radii PS and PS'. Equivalently, the normal at P bisects the interior angle \angle SPS'.

What this means physically. A ray of light travelling toward one focus of a hyperbolic mirror will be reflected so that it appears to come from the other focus.

Proof. Take the hyperbola \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 with foci S(ae, 0) and S'(-ae, 0). Let P = (x_1, y_1) be on the hyperbola. The tangent at P has the equation \dfrac{xx_1}{a^2} - \dfrac{yy_1}{b^2} = 1.

The tangent meets the x-axis at T = (a^2/x_1, 0). Now compute the ratio in which T divides SS' externally.

From the focal distance property of the hyperbola, SP = |ex_1 - a| and S'P = ex_1 + a (taking P on the right branch, so x_1 > 0 and ex_1 > a). Therefore:

The negative sign means T divides SS' externally in the ratio SP : S'P. By the angle bisector theorem (external division), the line PT — the tangent — is the external bisector of \angle SPS'. Therefore, the normal at P is the internal bisector.

This is the same result that holds for ellipses (where the tangent bisects the exterior angle and the normal bisects the interior angle), with one crucial difference: for an ellipse, a ray heading toward one focus reflects to the other focus. For a hyperbola, a ray heading toward one focus reflects away from the other focus — but along the line through the other focus.

Cassegrain telescopes exploit this. A parabolic primary mirror focuses starlight toward its focus. A small hyperbolic secondary mirror, placed so that one of its foci coincides with the parabolic focus, reflects the converging light back through a hole in the primary, toward the hyperbola's other focus where the eyepiece sits.

This property is not just a theoretical curiosity. ISRO's satellite communication systems and radio telescope designs use reflectors shaped as conic sections — parabolas for the primary dish and hyperbolas for the secondary reflector — precisely because of these angle-bisection properties.

The interplay between the three formulas

Before moving to examples, pause and notice how the three results — T = 0 for chord of contact, T = S_1 for midpoint chord, and SS_1 = T^2 for pair of tangents — are three faces of the same algebraic coin. All three come from substituting a line into the conic equation to get a quadratic, and then imposing a condition on that quadratic:

• Tangent (T = 0): the quadratic has a double root (the line touches the conic).
• Chord of contact (T = 0): the equation of the chord joining the two tangent contact points, derived from the tangent condition at each contact point.
• Midpoint chord (T = S_1): the sum of the roots of the quadratic is fixed (the midpoint is prescribed).
• Pair of tangents (SS_1 = T^2): the discriminant of the quadratic in t is zero (the line from the external point is tangent).

In each case, the substitution x^2 \to xx_1, y^2 \to yy_1 (the T operation) appears because it naturally arises when you take the difference of the conic equation at two points, or when you impose the double-root condition. The algebraic structure is the same — only the geometric question changes.

Worked examples

Common confusions

• "T = 0 always gives a tangent." Only when (x_1, y_1) is on the curve. When the point is external, T = 0 gives the chord of contact. When the point is internal, T = 0 gives the polar line (a concept from projective geometry), which does not directly correspond to tangent lines.

• "The chord of contact always crosses the hyperbola at two visible points." Not always. When the external point sees one tangent to each branch, the chord of contact joins points on different branches. In Cartesian terms, the chord line may not appear to "cut" the curve in a simple way — it passes between the branches.

• "SS_1 = T^2 gives me two separate lines." It gives the combined equation — a single second-degree equation whose graph is two lines. To get the individual lines, you would need to factor the equation, which is usually not required. Most problems ask for the combined equation itself.

• "T = S_1 works only for hyperbolas." It works for every conic. For a circle x^2 + y^2 = r^2, the chord with midpoint (h, k) is hx + ky = h^2 + k^2. For an ellipse, the same pattern. The T = S_1 formula is universal.

• "The reflection property says light reflects from one focus to the other." For an ellipse, yes — light from one focus reflects to the other focus. For a hyperbola, a ray aimed toward one focus reflects away from the other focus, along the line through it. The distinction matters in telescope design.

Going deeper

The pole and polar — the unifying idea

All four constructions in this article — tangent, chord of contact, chord with midpoint, pair of tangents — are shadows of a single deeper concept: the pole-polar correspondence.

Given a conic S = 0 and a point P = (x_1, y_1) (called the pole), the polar of P with respect to the conic is the line T = 0:

This is the same equation in all cases. What changes is the position of P:

Position of P	S_1	What T = 0 represents
On the conic	= 0	Tangent at P
Outside the conic	> 0	Chord of contact from P
Inside the conic	< 0	Polar line (no direct tangent interpretation)

The pole-polar relationship has a remarkable property called reciprocity: if the polar of P passes through Q, then the polar of Q passes through P. To see why, suppose P = (x_1, y_1) and Q = (x_2, y_2). The polar of P is:

If Q lies on this line, then:

But this expression is symmetric in the roles of (x_1, y_1) and (x_2, y_2) — it is the same as saying that P lies on the polar of Q. The symmetry of the bilinear expression T is doing all the work.

This reciprocity is the foundation of projective geometry and duality in conics. It means that the relationship between a point and its polar line is two-way — there is no preferred direction.

The director circle

The locus of all external points from which the two tangents to the hyperbola are perpendicular to each other is a circle called the director circle. For the hyperbola \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1, the director circle is:

This circle exists only when a > b. When a = b (a rectangular hyperbola), the director circle degenerates to a point at the origin. When a < b, there is no real locus — no external point gives perpendicular tangents.

Derivation. The two tangents from (x_1, y_1) in slope form are y_1 = mx_1 \pm \sqrt{a^2 m^2 - b^2}. Rearranging and squaring: (y_1 - mx_1)^2 = a^2 m^2 - b^2. This gives a quadratic in m:

The two roots m_1, m_2 are the slopes of the two tangents. For perpendicular tangents, m_1 m_2 = -1:

For the ellipse, the corresponding result is x^2 + y^2 = a^2 + b^2 — again, the sign difference between ellipse and hyperbola.

The number of tangents from a point

The sign of S_1 determines how many tangent lines can be drawn from (x_1, y_1) to the hyperbola:

• S_1 > 0 (exterior): exactly two tangents.
• S_1 = 0 (on the curve): exactly one tangent (the tangent at that point).
• S_1 < 0 (interior, between the branches): zero tangents — no line through this point can touch the curve without crossing it.

This is identical to the situation for ellipses and circles, and the same S_1 test works for all conics.

Comparing across conics — a summary table

Here is the complete comparison of the advanced formulas across all three central conics:

Formula	Circle x^2 + y^2 = r^2	Ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1	Hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
Tangent at point	xx_1 + yy_1 = r^2	\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1	\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1
Chord of contact	Same as tangent formula	Same as tangent formula	Same as tangent formula
Midpoint chord	T = S_1	T = S_1	T = S_1
Pair of tangents	SS_1 = T^2	SS_1 = T^2	SS_1 = T^2
Director circle	x^2+y^2=2r^2	x^2+y^2=a^2+b^2	x^2+y^2=a^2-b^2

The first four rows are identical in structure — the only thing that changes is the definition of S, S_1, and T for each conic. The director circle is the one place where the sign difference between ellipse and hyperbola produces a qualitatively different result (the hyperbola's may not exist as a real circle).

This universality is why the S-T-S_1 notation is so powerful. Learn it once, and it works for every conic you will ever meet.

Where this leads next

You now have the full advanced toolkit for hyperbolas. The natural continuations:

• Hyperbola — Tangent and Normal — the tangent and normal formulas that this article builds on.
• Hyperbola — Introduction — the definition, foci, eccentricity, and asymptotes.
• Ellipse — Advanced — the same constructions (chord of contact, pair of tangents, reflection) for the ellipse.
• Parabola — Advanced — the same constructions for the parabola.
• Pair of Tangents and Chord of Contact — the unified treatment across all conics.