equation of tangent of circle

We derive the equation of tangent line for a circle with radius r.  For simplicity, we chose for the origin the centre of the circle, when the points  (x,y)  of the circle satisfy the equation

x2+y2=r2. (1)

Let the point of tangency be  (x0,y0).  Then the slope of radius with end pointPlanetmathPlanetmath(x0,y0)  is y0x0, whence, according to the parent entry (http://planetmath.org/TangentOfCircle), its opposite inverse -x0y0 is the slope of the tangent, being perpendicularMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the radius.  Thus the equation of the tangent is written as


Removing the denominator and the parentheses we obtain from this first  x0x+y0y=x02+y02,  and then

x0x+y0y=r2 (2)

since  (x0,y0)  satisfies (1).

Remark.  In the equation (2) of the tangent, x0, y0 are the coordinatesMathworldPlanetmathPlanetmath of the point of tangency and x,y the coordinates of an arbitrary point of the tangent line.  But one can of course swap those meanings; then we interprete (2) such that x0, y0 are the coordinates of some point P outside the circle (1) and x,y the coordinates of the point of tangency of either of the tangents which may be drawn from P to the circle.  If (2) now is again interpreted as an equation of a line (its degree (http://planetmath.org/AlgebraicEquation) is 1!), this line must pass through both the mentioned points of tangency A and B (they satisfy the equation!); in a , (2) is now the equation of the tangent chord AB of  P=(x0,y0).  See also http://mathworld.wolfram.com/Polar.htmlpolar.

Title equation of tangent of circle
Canonical name EquationOfTangentOfCircle
Date of creation 2013-03-22 18:32:16
Last modified on 2013-03-22 18:32:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Derivation
Classification msc 51M20
Classification msc 51M04
Related topic Polarising