Tangent Line to a Circle Calculator

Find the equations of tangent lines from an external point to a circle.

Circle & External Point

Result

Tangent Line 1
--
Tangent Line 2
--
Circle center--
Radius--
External point--
Distance to center--
Tangent length--

Step-by-Step Solution

(x-h)^2 + (y-k)^2 = r^2

Tangent Lines to a Circle

A tangent line to a circle is a line that touches the circle at exactly one point. From an external point, exactly two tangent lines can be drawn to a circle. The tangent line is always perpendicular to the radius at the point of tangency.

Key Concepts

Tangent Length

The length of the tangent from external point P to the circle equals sqrt(d^2 - r^2) where d is the distance from P to the center.

L = sqrt(d^2 - r^2)

Perpendicularity

The tangent line is always perpendicular to the radius drawn to the point of tangency.

radius _|_ tangent

Equal Tangent Lengths

Both tangent lines from the same external point have equal length.

|PT1| = |PT2|

Method

To find tangent lines from an external point (x0, y0) to a circle with center (h, k) and radius r, we find lines through (x0, y0) whose distance from the center equals r. This leads to a quadratic equation in the slope m, yielding two tangent lines (or one if the point is on the circle).

Special Cases

  • If the point is inside the circle, no real tangent lines exist.
  • If the point is on the circle, there is exactly one tangent line.
  • If the point is outside the circle, there are exactly two tangent lines.