Find the tangent line equation from an external point to a circle
Center X (h)
Center Y (k)
Radius (R)
Point X (x0)
Point Y (y0)
Result
Tangent Status
-
Tangent Length
-
Tangent 1 Slope
-
Tangent 1 Equation
-
Detailed Derivation
Tangent Line Formulas
Distance d = sqrt((x0-h)^2 + (y0-k)^2)
Tangent length t = sqrt(d^2 - R^2)
Slope of radius m_r = (y0-k)/(x0-h)
Tangent slopes: m_t satisfy m_r x m_t = -1
A tangent line touches a circle at exactly one point and is perpendicular to the radius at that point of contact. This fundamental property is used in circle geometry proofs and real-world applications.
⚠If the point is inside the circle (d < R), no real tangents exist. If on the circle (d = R), one tangent. If outside (d > R), two tangents.
What Is a Tangent to a Circle?
A tangent is a line that touches a circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent. This perpendicular relationship is the key to finding tangent equations.
Perpendicular Property
The tangent is perpendicular to the radius at the point of contact. Product of slopes = -1.
Two Tangents
From any external point, two equal-length tangents can be drawn. They are symmetric about the line joining the point to the center.
Tangent Length
Length t = sqrt(d^2 - R^2) by the Pythagorean theorem, where d is the distance from point to center.
Applications
Tangents are used in optics (reflection/refraction), gear design, robot path planning, and geometry proofs.
Teaching Example: Circle center O(0,0), R=5, external point P(8,6). d = sqrt(64+36) = 10. Tangent length = sqrt(100-25) = 8.66. The radius OP has slope = 6/8 = 0.75, so tangent slopes are -4/3.
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