Tangent and Normal to a Hyperbola: Formulas with Examples

A tangent to a hyperbola is a straight line that touches the curve at exactly one point and runs in the same direction as the curve there. A normal is the line through that same point which is perpendicular to the tangent. Together they capture the curve's direction and turning at any chosen point.

Tangents and normals matter far beyond exams. The reflecting property of the hyperbola, a ray aimed at one focus bounces off toward the other, is proved using the tangent, and engineers use it to design hyperbolic mirrors, telescope optics, and navigation systems. In calculus, the tangent's slope is the derivative, the starting point of all rates of change.

Key Concepts

• Point of contact (x₁, y₁) must satisfy x₁²/a² − y₁²/b² = 1.
• Tangent at (x₁, y₁): xx₁/a² − yy₁/b² = 1, obtained by replacing x² with xx₁ and y² with yy₁ (the T = 0 rule).
• Normal at (x₁, y₁): a²x/x₁ + b²y/y₁ = a² + b².
• Slope of tangent: differentiating gives dy/dx = b²x/(a²y) at any point (x, y) on the curve.
• Touch condition: a line y = mx + c touches the hyperbola at some point exactly when c² = a²m² − b².
• Chord of contact: from an outside point (x₁, y₁), the line T = 0 with T = xx₁/a² − yy₁/b² − 1 joins the two contact points of the tangents drawn from it.

Formulas

• Tangent: xx₁/a² − yy₁/b² = 1. Here (x₁, y₁) is the point of contact, (x, y) is any running point on the line, and a, b are the semi-axes (length units).
• Normal: a²x/x₁ + b²y/y₁ = a² + b². Assumption: x₁ ≠ 0 and y₁ ≠ 0; at a vertex the normal is simply the x-axis.
• Touch condition: for y = mx + c, c² = a²m² − b², with m dimensionless; real contact needs a²m² − b² ≥ 0.
• Limits: the tangent and normal formulas describe one specific point; the touch condition describes a whole line but not where it touches.

Worked Example

Find the tangent and normal to x²/25 − y²/16 = 1 at the point (5√2, 4). Step 1: check the point lies on the curve: (5√2)²/25 = 50/25 = 2, 4²/16 = 1, and 2 − 1 = 1. Step 2: tangent: x(5√2)/25 − y(4)/16 = 1, which simplifies to (√2/5)x − y/4 = 1, or 4√2x − 5y = 20. Step 3: normal: 25x/(5√2) + 16y/4 = 25 + 16 = 41, i.e. (5/√2)x + 4y = 41, or 5x + 4√2y = 41√2. Step 4: check perpendicularity: slopes 4√2/5 and −5/(4√2) multiply to −1. Step 5: touch test: does y = 2x + 3 touch this hyperbola? c² = 9 but a²m² − b² = 100 − 16 = 84, and 9 ≠ 84, so it does not. The line y = 2x + √84 = 2x + 2√21 would touch.

Common Misconceptions

• "The tangent formula works for any point you plug in." It gives the tangent only for a point on the hyperbola; for an outside point it gives the chord of contact instead.
• "The normal formula works everywhere on the curve." At a vertex like (a, 0), y₁ = 0 causes division by zero; the normal there is just the x-axis, y = 0, handled as a special case.
• "c² = a²m² − b² gives the tangent at a point." It only says whether a line of slope m touches somewhere; for a specific point use xx₁/a² − yy₁/b² = 1.

Real-World Applications

• Hyperbolic mirrors in Cassegrain telescopes use the tangent-based reflection property to focus light from one focus to the other.
• The LORAN navigation system located ships and aircraft from intersecting hyperbolas, with tangents giving signal directions.
• Cooling towers and arches with hyperbolic profiles need tangent calculations for their structural design.

For x²/a² − y²/b² = 1, the tangent at (x₁, y₁) is xx₁/a² − yy₁/b² = 1, the normal is a²x/x₁ + b²y/y₁ = a² + b², and y = mx + c touches exactly when c² = a²m² − b². One point on the curve unlocks the whole local geometry.