Chord of Contact: Explained with Examples, Notes and FAQs

The chord of contact is the straight line joining the two points where the tangents from an external point touch a circle. If P = (x₁, y₁) lies outside the circle x² + y² + 2gx + 2fy + c = 0, the chord of contact is given by the same T = 0 formula as the tangent: xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0 — but here it joins the two contact points instead of touching at one.

This is useful because it finds the contact points without solving for tangent slopes: intersect the chord of contact with the circle and you get both contact points at once. It is also the polar of an external point, a concept that extends to all conics. Exam problems often ask for the chord of contact's equation, its length, or its midpoint, and the T = 0 shortcut answers the first instantly.

Key Concepts

• Equation: T = 0, i.e. xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0, where (x₁, y₁) is the external point.
• Why it works: each contact point A satisfies both the circle and the relation x_Ax₁ + y_Ay₁ + … = 0, so both lie on the line T = 0.
• Position requirement: S₁ > 0 (point outside); for S₁ = 0 the "chord" shrinks to the tangent at the point itself.
• Midpoint: the chord's midpoint lies on the line joining the external point to the centre.

Formulas

• Chord of contact from (x₁, y₁): xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0.
• Length of the chord: with tangent length L = sqrt(S₁) and radius r, chord length = 2Lr / sqrt(L² + r²).
• Units: all terms are areas, consistent throughout.
• Assumption: the point is external (S₁ > 0); the two contact points are real and distinct.

Worked Example

Find the chord of contact of (7, 1) with respect to x² + y² = 25, and verify it passes through the contact points.

1. Apply T = 0 with (x₁, y₁) = (7, 1): 7x + y − 25 = 0.
2. Find contact points by intersecting with the circle: y = 25 − 7x gives x² + (25 − 7x)² = 25, i.e. x² − 7x + 12 = 0, so x = 3 or 4.
3. Contact points: (3, 4) and (4, −3).
4. Check both on 7x + y = 25: 21 + 4 = 25 and 28 − 3 = 25. Both satisfy it, confirming the line joins the contact points.
5. Chord length check: L = 5, r = 5, so chord = 2·5·5/sqrt(50) = 50/(5·sqrt(2)) = 5·sqrt(2) ≈ 7.07. The distance between (3, 4) and (4, −3) is sqrt(1 + 49) = sqrt(50) ≈ 7.07. Matches.

Common Misconceptions

• "T = 0 from an external point is a tangent." It is the chord joining the two contact points — the tangents themselves need the separate pair-of-tangents equation.
• "The chord of contact passes through the external point." It does not: (7, 1) gives 49 + 1 = 50 ≠ 25.
• "You must find the contact points first." The T = 0 formula gives the chord directly; contact points come afterwards by intersection.

Real-World Applications

• Optics: the chord of contact marks where tangent rays from a point source graze a circular mirror or lens.
• Robotics: tangent-point geometry plans the shortest path that grazes a circular obstacle — the chord of contact locates the touch points.
• Cartography: lines of sight tangent to a circular region of interest are computed through pole–polar relations.
• Gear geometry: common tangent points between pitch circles are found via chord-of-contact style equations.

In short: from an external point, T = 0 draws the chord joining the two tangent contact points — the same formula as the tangent, reinterpreted. Use it to find contact points by intersection, check the point is genuinely outside first, and never mistake this chord for a tangent line.