Tangent and Normal to an Ellipse: Formulas with Examples

A tangent to an ellipse is a line that touches it at exactly one point, grazing the curve without crossing it. The normal at the same point passes through the point of contact perpendicular to the tangent. For x²/a² + y²/b² = 1, the tangent at (x₁, y₁) on the ellipse is xx₁/a² + yy₁/b² = 1, and the normal is a²x/x₁ − b²y/y₁ = a² − b².

These lines matter wherever curves meet the straight-line world: light reflecting off elliptical mirrors, balls bouncing off elliptical walls, planets moving along orbits — the tangent gives the direction of motion, the normal the direction of the push. They are also exam favourites, since a few short formulas answer most questions.

Key Concepts

• Tangent at (x₁, y₁): xx₁/a² + yy₁/b² = 1, valid only when the point lies on the ellipse.
• Normal at (x₁, y₁): a²x/x₁ − b²y/y₁ = a² − b², perpendicular to the tangent.
• T notation: T = xx₁/a² + yy₁/b² − 1; the tangent is T = 0.
• Chord of contact: from an external point (x₁, y₁), the chord joining the two contact points is also T = 0.
• Touch condition: y = mx + c touches the ellipse iff c² = a²m² + b².

Formulas

• Tangent: xx₁/a² + yy₁/b² = 1.
• Normal: a²x/x₁ − b²y/y₁ = a² − b².
• Chord of contact from external (x₁, y₁): xx₁/a² + yy₁/b² = 1.
• Touch test: c² = a²m² + b² for the line y = mx + c.
• Units: lengths for x, y, a, b, c; slope m is dimensionless. The point must satisfy the ellipse equation.

Worked Example

Find the tangent and normal to x²/25 + y²/9 = 1 at (3, 12/5).

1. Verify the point. 9/25 + (144/25)/9 = 9/25 + 16/25 = 1. It lies on the ellipse.
2. Tangent. 3x/25 + (12/5)y/9 = 1 → 3x/25 + 12y/45 = 1 → multiply by 225: 27x + 60y = 225 → 9x + 20y = 75. Check: 27 + 48 = 75.
3. Normal. 25x/3 − 9y/(12/5) = 16 → 25x/3 − 45y/12 = 16 → multiply by 12: 100x − 45y = 192. Check: 300 − 108 = 192.

Answer: tangent 9x + 20y = 75, normal 100x − 45y = 192.

Common Misconceptions

• "The tangent formula works for any point." No — the point must lie on the ellipse; for an outside point, T = 0 gives the chord of contact instead.
• "The normal is the tangent with x and y swapped." No — it has its own form, a²x/x₁ − b²y/y₁ = a² − b².
• "You must solve a quadratic to test tangency." The condition c² = a²m² + b² answers directly.

Real-World Applications

• Optics: the tangent at each point of an elliptical mirror sets the angle at which light reflects from one focus to the other.
• Orbits: the tangent at a planet's position gives its instantaneous direction of travel.
• Computer graphics: bounce direction off an elliptical boundary is computed from the normal at the impact point.
• Road design: tangent lines to elliptical curves plan smooth entries and exits on curved roads.

In short: tangent xx₁/a² + yy₁/b² = 1, normal a²x/x₁ − b²y/y₁ = a² − b², touch test c² = a²m² + b². Check the point is on the ellipse first.