Circle Tangent Calculator

Find tangent lines to a circle from a point on the circle or an external point. Step-by-step solutions with interactive graph.

Circle equation: (x - h)² + (y - k)² = r²

Tangent line from a point (x₀, y₀) on the circle: perpendicular to radius.

From an external point: two tangents exist if point is outside the circle.

Adjusts point to lie exactly on the circle (for 'on circle' mode).
Calculating...

Understanding Circle Tangents

A tangent line to a circle touches the circle at exactly one point, called the point of tangency. The radius drawn to the point of tangency is always perpendicular to the tangent line.

Key properties:

  • If a point lies on the circle, there is exactly one tangent line.
  • If a point is outside the circle, there are exactly two tangent lines to the circle from that point.
  • The distance from the center to the tangent line equals the radius.
  • The angle between two tangents from an external point is related to the central angle.

Methods of Calculation

Point on the circle (x₀, y₀)

The tangent line is perpendicular to the radius vector (x₀ - h, y₀ - k). Its equation can be written as:

(x₀ - h)(x - x₀) + (y₀ - k)(y - y₀) = 0

Or in slope form: slope m = -(x₀ - h)/(y₀ - k) (if denominator non-zero).

External point (x₀, y₀)

Let the tangent line have equation y = m(x - x₀) + y₀. Substitute into circle equation and set discriminant = 0 to solve for m. Alternatively, use the formula for length of tangent and the chord of contact.

(y₀ - k) = m(x₀ - h) ± r √(1 + m²)

Solving gives two slopes (if they exist).

Applications

  • Geometry: Constructing tangents to circles.
  • Physics: Reflection off circular mirrors.
  • Engineering: Gear design, cam mechanisms.
  • Computer Graphics: Collision detection, path planning.

Calculator features: Symbolic algebra using math.js, interactive graph, step-by-step results, handles both point types and special cases (vertical tangents).

Frequently Asked Questions

If the point is inside the circle (distance from center < radius), no real tangent lines exist. The calculator will show an error message.

Yes. For example, at the far right point of a circle centered at origin, the tangent is vertical. The calculator handles vertical lines by displaying equations in the form x = constant.

They are usually labeled as "upper" and "lower" or "left" and "right" depending on slope. The graph will show both in different colors.