Hyperbola — Tangent and Normal

A satellite dish is shaped as a paraboloid, but the receiver at its focal point often uses a small hyperbolic reflector to redirect the signal. The reflector works because of a specific geometric property: a ray aimed at one focus of a hyperbola bounces off the curve as though it came from the other focus. To design that reflector — to know exactly where the reflected ray goes — you need the tangent line at each point of the hyperbola. The tangent tells you the surface direction, and the surface direction tells you the reflection angle.

Take the hyperbola \dfrac{x^2}{9} - \dfrac{y^2}{4} = 1. Pick a point on it — say (3\sec 60°,\, 2\tan 60°) = (6, 2\sqrt{3}). Verify: \dfrac{36}{9} - \dfrac{12}{4} = 4 - 3 = 1. Good, the point sits on the curve.

Now ask: what line just grazes the hyperbola at this point? Not a secant that cuts through and exits the other side — a line that touches and then peels away, sharing only that one contact point in its neighbourhood. That line is the tangent.

And once you have the tangent, the line perpendicular to it through the same point is the normal — the direction pointing straight into (or away from) the curve. The normal is what matters for reflection: a ray bounces symmetrically about the normal.

For a circle, tangent lines are straightforward: every tangent is perpendicular to the radius. For a hyperbola, there is no single centre-to-point radius that works so neatly, so you need calculus or algebraic tricks. The results, though, are just as clean — and the condition for when a line can be tangent is one of the most elegant equations in coordinate geometry.

The idea before the formula

Before jumping into derivations, picture what is happening. A tangent line to any curve at a point P is the line that best approximates the curve near P. If you zoom in on the curve at P far enough, the curve and the tangent line become indistinguishable — they merge into the same straight line.

For the hyperbola, the tangent at a vertex (say (a, 0)) is vertical — the line x = a. As you move along the curve away from the vertex, the tangent rotates, becoming less and less steep. Far from the centre, the tangent approaches the asymptote direction but never quite reaches it — the tangent always makes a slightly different angle from the asymptote.

The normal, being perpendicular to the tangent, starts horizontal at the vertex and rotates in the opposite direction. At the vertex (a, 0), the normal is the x-axis itself — the transverse axis.

The tangent at a point — the calculus route

Start with the standard hyperbola \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1. You want the slope of the curve at a point (x_1, y_1) on it.

Differentiate both sides implicitly with respect to x:

Solve for \dfrac{dy}{dx}:

At the point (x_1, y_1), the slope is m = \dfrac{b^2 x_1}{a^2 y_1}.

The tangent line through (x_1, y_1) with this slope is:

Multiply both sides by a^2 y_1:

Rearrange:

Divide both sides by a^2 b^2:

But (x_1, y_1) lies on the hyperbola, so \dfrac{x_1^2}{a^2} - \dfrac{y_1^2}{b^2} = 1. The right side is just 1.

Notice the pattern. The hyperbola equation has x^2 and y^2. In the tangent equation, one copy of x is replaced by x_1 and one copy of y by y_1. This same substitution rule — sometimes called the T = S₁ trick — works for circles, parabolas, and ellipses too. It is worth memorising once and using everywhere.

Tangent in parametric form

A point on the hyperbola can be written as (a\sec\theta, b\tan\theta), where \theta is the parameter from the parametric representation.

Substitute x_1 = a\sec\theta and y_1 = b\tan\theta into the Cartesian tangent equation:

This form is especially useful when you are given the eccentric angle \theta rather than explicit coordinates.

Tangent in slope form — and the condition for tangency

Here is the question that matters most in problems: given a line y = mx + c, when is it tangent to the hyperbola?

Substitute y = mx + c into \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1:

Multiply through by a^2 b^2:

This is a quadratic in x. A tangent line touches the hyperbola at exactly one point, so this quadratic must have exactly one solution — meaning its discriminant is zero.

The discriminant is:

Set D = 0 (and divide out the 4a^2):

Expand the second term:

The a^2 m^2 c^2 terms cancel:

Divide by b^2:

Compare this with the ellipse, where the condition is c^2 = a^2 m^2 + b^2 — a plus sign instead of a minus. That single sign change reflects the fundamental difference between the two curves. For the ellipse, every slope has a tangent (you can always touch an ellipse from any direction). For the hyperbola, slopes with |m| < b/a have no tangent at all — those lines would need to pass through the gap between the two branches.

The tangent line in slope form is therefore:

For each valid slope m, there are exactly two tangent lines — one to each branch of the hyperbola. The + sign gives the tangent that touches the branch in the upper half-plane (or the right branch, depending on the slope), and the - sign gives the tangent to the other branch.

What happens at the boundary? When |m| = b/a exactly, the condition becomes c^2 = 0, so c = 0 and the tangent passes through the origin. The line y = (b/a)x through the origin is the asymptote — you can think of the asymptote as a "tangent at infinity." As the contact point moves farther and farther along a branch of the hyperbola, the tangent rotates toward the asymptote direction and the intercept c shrinks toward zero.

The equation of the normal

The normal at a point is the line perpendicular to the tangent at that point. Since the tangent has slope \dfrac{b^2 x_1}{a^2 y_1}, the normal has slope -\dfrac{a^2 y_1}{b^2 x_1} (the negative reciprocal).

The normal through (x_1, y_1):

Multiply through by b^2 x_1:

Dividing both sides by x_1 y_1:

In parametric form, substituting x_1 = a\sec\theta, y_1 = b\tan\theta:

Properties of tangent and normal

Several geometric properties follow from the equations above. Each one is worth knowing — they appear frequently in problems.

1. The tangent at an endpoint of the latus rectum. The latus rectum endpoints are at (\pm ae, \pm b^2/a). Taking the point (ae, b^2/a) and using the tangent formula:

So the tangent at the end of the latus rectum has slope e — the eccentricity of the hyperbola.

2. The tangent intercepts. The tangent \dfrac{x x_1}{a^2} - \dfrac{y y_1}{b^2} = 1 meets the x-axis (set y = 0) at x = a^2/x_1 and the y-axis (set x = 0) at y = -b^2/y_1.

3. The product of perpendicular distances from the foci. If p_1 and p_2 are the perpendicular distances from the two foci (\pm ae, 0) to any tangent line, then p_1 \cdot p_2 = b^2. This is a constant — it does not depend on which tangent you pick.

To prove this, take the slope-form tangent mx - y + c = 0 where c^2 = a^2 m^2 - b^2. The distances from the foci (ae, 0) and (-ae, 0) are:

Since b^2 = a^2(e^2 - 1), substitute:

4. The normal bisects the angle between the focal radii. If you draw lines from the two foci to a point on the hyperbola, the normal at that point bisects the angle between these two focal radii. This is the reflection property — it is the reason hyperbolic mirrors are used in telescopes (more on this in the Hyperbola — Advanced article).

5. Tangent subtends supplementary angles at the foci. If the tangent at any point P on the hyperbola meets the directrices or is viewed from the foci, the tangent line subtends equal angles at the two foci — specifically, if S and S' are the foci and T is the tangent line at P, the angles \angle SPT and \angle S'PT are supplementary.

6. The foot of the perpendicular from a focus lies on the auxiliary circle. Drop a perpendicular from a focus to any tangent line. The foot of this perpendicular lies on the auxiliary circle x^2 + y^2 = a^2. This can be proved by finding the foot of perpendicular from (ae, 0) to the tangent y = mx + \sqrt{a^2 m^2 - b^2} and showing that its coordinates satisfy x^2 + y^2 = a^2.

These properties are not isolated curiosities — they connect the tangent theory to the geometry of the foci, the directrices, and the auxiliary circle, giving you multiple ways to attack the same problem.

Worked examples

Common confusions

• "The tangent touches the hyperbola at one point, so it crosses only once." This is true locally — near the point of tangency — but the tangent line extends to infinity and may actually cross the other branch of the hyperbola at a second point. For instance, a tangent at a point near the vertex of one branch can cross the other branch far away. "Tangent" means local contact, not global.

• "The condition c^2 = a^2 m^2 - b^2 has a plus sign, like the ellipse." The ellipse condition is c^2 = a^2 m^2 + b^2 — with a plus. Mixing up the sign is the most common error in conic tangent problems. Remember: the hyperbola equation has a minus sign (x^2/a^2 - y^2/b^2), and so does its tangency condition (a^2 m^2 - b^2). The signs match.

• "Every slope gives a tangent." Only slopes with |m| \geq b/a produce real tangent lines. For the hyperbola \dfrac{x^2}{9} - \dfrac{y^2}{4} = 1, you need |m| \geq 2/3. Slopes flatter than this correspond to lines that slip between the two branches. The boundary case |m| = b/a gives the asymptotic direction — the tangent "at infinity."

• "The normal formula has a minus sign between the two terms." The tangent has a minus sign (xx_1/a^2 - yy_1/b^2 = 1), but the normal has a plus sign (a^2 x/x_1 + b^2 y/y_1 = a^2 + b^2). The perpendicularity flips the sign.

Going deeper

The reflection property — proved

Here is the property that gives hyperbolas their role in optics: the tangent at any point P on the hyperbola makes equal angles with the two focal radii PS and PS'.

Equivalently, the normal at P bisects the angle \angle SPS', where S and S' are the two foci.

To prove this, take P = (a\sec\theta, b\tan\theta) and the foci S = (ae, 0), S' = (-ae, 0).

The tangent at P is \dfrac{x\sec\theta}{a} - \dfrac{y\tan\theta}{b} = 1, which in the form Ax + By + C = 0 is:

The tangent of the angle \alpha_1 that the tangent line makes with PS can be computed using the formula for the angle between two lines. After a careful computation (using b^2 = a^2(e^2 - 1) and the focal distance formulas SP = |ex_1 - a| and S'P = |ex_1 + a|), both angles turn out to be equal. The key identity that makes the algebra collapse is:

This means a ray of light aimed at one focus of a hyperbolic mirror will reflect as though it came from the other focus. Cassegrain telescopes use exactly this: a hyperbolic secondary mirror redirects light from the parabolic primary mirror through a hole in its centre, because both foci of the hyperbola are aligned with the parabolic focus and the eyepiece.

The auxiliary circle and the tangent

If you draw the auxiliary circle x^2 + y^2 = a^2 (the circle on the transverse axis as diameter), there is a clean connection between the tangent at a point P on the hyperbola and the tangent at the corresponding point Q on the auxiliary circle — the point Q = (a\cos\theta, a\sin\theta) that shares the same eccentric angle \theta as P = (a\sec\theta, b\tan\theta).

The hyperbola tangent at P is \dfrac{x\sec\theta}{a} - \dfrac{y\tan\theta}{b} = 1. Setting y = 0, the tangent meets the x-axis at x = a\cos\theta.

The auxiliary circle tangent at Q is x\cos\theta + y\sin\theta = a. Setting y = 0, this meets the x-axis at x = a/\cos\theta = a\sec\theta.

So the x-intercepts are different — a\cos\theta for the hyperbola tangent and a\sec\theta for the circle tangent. But the normals at these corresponding points do meet on the x-axis at a common point, and this relationship is useful in geometric constructions and locus problems involving conics and their auxiliary circles.

The length of the tangent and normal segments

Two useful lengths can be read from the geometry:

• Tangent length (from P to the x-axis along the tangent): the tangent at P = (x_1, y_1) meets the x-axis at (a^2/x_1, 0). The distance from P to this intercept is the tangent length.

• Sub-tangent (the projection of the tangent segment onto the x-axis): this is |x_1 - a^2/x_1| = |x_1^2 - a^2|/|x_1|.

• Sub-normal (the projection of the normal segment onto the x-axis): from the normal equation, the normal meets the x-axis at x = (a^2 + b^2)x_1 / a^2 = e^2 x_1 (since a^2 + b^2 = a^2 e^2 for a hyperbola). So the sub-normal is |e^2 x_1 - x_1| = (e^2 - 1)|x_1| = (b^2/a^2)|x_1|.

The sub-normal being proportional to x_1 is a distinctive property of the hyperbola. For a parabola, the sub-normal is a constant (independent of the point).

Normal in slope form

If the normal has slope m, the foot of the normal on the hyperbola can be expressed in terms of m. From the parametric normal ax\cos\theta + by\cot\theta = a^2 + b^2, the slope of the normal is:

So \sin\theta = -bm_n/a, and from \sin^2\theta + \cos^2\theta = 1:

For \cos\theta to be real, you need |m_n| < a/b. This is the condition for a normal with slope m_n to exist.

Substituting back into the parametric normal equation gives:

This form is useful when you need to find normals of a specified slope, just as the slope form of the tangent is useful for tangents. The \mp gives two normals for each valid slope — one on each branch. The condition |m_n| < a/b is the complement of the tangent condition |m| \geq b/a: slopes that are too flat for a tangent are exactly the slopes that are steep enough for a normal, and vice versa.

Comparing tangent equations across conics

It is worth pausing to see the tangent equations for all three conics side by side. The pattern is striking:

Conic	Equation	Tangent at (x_1, y_1)	Slope-form condition
Circle	x^2 + y^2 = a^2	xx_1 + yy_1 = a^2	c^2 = a^2(1 + m^2)
Ellipse	\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1	\dfrac{xx_1}{a^2} + \dfrac{yy_1}{b^2} = 1	c^2 = a^2 m^2 + b^2
Hyperbola	\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1	\dfrac{xx_1}{a^2} - \dfrac{yy_1}{b^2} = 1	c^2 = a^2 m^2 - b^2

The rule is mechanical: to get the tangent equation, replace x^2 \to xx_1 and y^2 \to yy_1 in the conic's equation. The sign pattern in the tangent matches the sign pattern in the conic. This is the T = 0 substitution, and it works because the tangent condition is equivalent to requiring a quadratic (from substituting the line into the conic) to have a repeated root.

Where this leads next

You now have the full toolkit for tangent and normal lines to a hyperbola. The next set of ideas builds on this:

• Hyperbola — Advanced — chord of contact, chord with a given midpoint, pair of tangents from an external point, and the reflection property applied.
• Hyperbola — Introduction — if you need to revisit the basics: definition, foci, eccentricity, asymptotes.
• Hyperbola — Parametric and Auxiliary Circle — the parametric form and its geometric meaning.
• Ellipse — Tangent and Normal — the sister article for ellipses, with the same structure but a plus sign where the hyperbola has a minus.
• Parabola — Tangent and Normal — tangent and normal for the other open conic.