Tangent to a Circle: Equation, Length Formula, Examples

A tangent to a circle is a line that touches the circle at exactly one point. If (x₁, y₁) lies on the circle x² + y² + 2gx + 2fy + c = 0, the tangent there is given by the elegant T = 0 formula: xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0. From an external point, two tangents can be drawn, and their length is sqrt(S₁), where S₁ is the circle's left-hand side evaluated at the point.

Tangents matter everywhere circles meet lines: wheels rolling on roads, light rays grazing a lens, cables leaving a pulley. The T = 0 rule also generalises to every conic, so learning it for circles teaches a pattern you will reuse for parabolas, ellipses and hyperbolas. Exam questions love the length-of-tangent formula because it tests the S₁ = 0 point-circle idea.

Key Concepts

• T = 0: for a point (x₁, y₁) ON the circle, xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0 is the tangent at that point.
• Length of tangent: from an external point (x₁, y₁), length = sqrt(S₁) where S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c.
• Perpendicularity: the tangent at a point is perpendicular to the radius drawn to that point.
• External vs internal: S₁ > 0 means the point is outside (real tangents exist); S₁ < 0 means inside (no real tangent).

Formulas

• Tangent at (x₁, y₁) on the circle: xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0.
• Tangent length from (x₁, y₁): L = sqrt(x₁² + y₁² + 2gx₁ + 2fy₁ + c).
• Units: L carries length units; the formula is dimensionally consistent since each term is an area.
• Assumption: (x₁, y₁) must actually lie on the circle for T = 0 to be a tangent; for an outside point it gives the chord of contact instead.

Worked Example

For the circle x² + y² = 25 and the external point P = (7, 1), find the tangent length, the points of contact, and one tangent equation.

1. Tangent length: S₁ = 49 + 1 − 25 = 25, so L = sqrt(25) = 5.
2. The chord of contact from P is T = 0 with (x₁, y₁) = (7, 1): 7x + y − 25 = 0, i.e. y = 25 − 7x. The contact points lie where this meets the circle.
3. Substitute: x² + (25 − 7x)² = 25 gives 50x² − 350x + 600 = 0, i.e. x² − 7x + 12 = 0, so x = 3 or x = 4.
4. Contact points: (3, 4) and (4, −3). Check (3, 4): 9 + 16 = 25, on the circle.
5. Tangent at (3, 4) by T = 0: 3x + 4y = 25. Verify perpendicularity: radius slope 4/3, tangent slope −3/4, product −1. Also the distance from (7, 1) to (3, 4) is sqrt(16 + 9) = 5, matching the tangent length.

Common Misconceptions

• "T = 0 always gives a tangent." Only when the point is on the circle; from an external point it gives the chord of contact, and from an internal point a meaningless line.
• "There is one tangent from an outside point." There are two, symmetric about the line joining the external point to the centre.
• "Tangent length is S₁ itself." It is sqrt(S₁) — S₁ is a squared length.

Real-World Applications

• Belts and pulleys: the free span of a belt between two pulleys lies along common external tangents of the two circles.
• Optics: rays tangent to a circular mirror graze it — the limiting case between hitting and missing.
• Surveying: sight lines tangent to a circular obstacle give the shortest visible path around it.
• Robotics: a robot arm's reach envelope is a circle, and tangent paths give collision-free grazes past circular obstacles.

In short: T = 0 writes the tangent at a circle point in one step, and sqrt(S₁) measures tangent length from outside. Check the point is on the circle before using T = 0 as a tangent, verify perpendicularity with the radius, and remember an external point always yields two symmetric tangents.