|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
equation of tangent of circle
|
(Derivation)
|
|
|
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
 |
(1) |
Let the point of tangency be $(x_0,\,y_0)$ . Then the slope of radius with end point $(x_0,\,y_0)$ is $\frac{y_0}{x_0}$ , whence, according to the parent entry, its opposite inverse $-\frac{x_0}{y_0}$ is the slope of the tangent, being perpendicular to the radius. Thus the equation of the tangent is
written as $$y-y_0 \;=\; -\frac{x_0}{y_0}(x-x_0).$$ Removing the denominator and the parentheses we obtain from this first $x_0x+y_0y = x_0^2+y_0^2$ , and then
 |
(2) |
since $(x_0,\,y_0)$ satisfies (1).
Remark. In the equation (2) of the tangent, $x_0$ , $y_0$ are the coordinates 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 $x_0$ , $y_0$ are the coordinates of some fixed 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 is 1!), this line must pass through both the mentioned points of tangency $A$ and $B$ (they satisfy the equation!); in a word, (2) is now the equation of the tangent chord $AB$ of $P = (x_0,\,y_0)$ . See also polar.
|
"equation of tangent of circle" is owned by pahio.
|
|
(view preamble | get metadata)
Cross-references: tangent chord, pass through, line, coordinates, denominator, perpendicular, tangent, opposite inverse, end point, slope, points, centre, origin, radius, circle, tangent line, equation
There is 1 reference to this entry.
This is version 7 of equation of tangent of circle, born on 2008-11-11, modified 2009-02-20.
Object id is 11254, canonical name is EquationOfTangentOfCircle.
Accessed 3252 times total.
Classification:
| AMS MSC: | 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries) | | | 51M20 (Geometry :: Real and complex geometry :: Polyhedra and polytopes; regular figures, division of spaces) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
(-3,-2.3335) \rput(-5.5,-3){.} \rput(5.5,3){.} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/11254/js/img3.png "\begin{pspicture}(-5.5,-3)(5.5,3) \rput(-2.85,-0.1){$O$} \rput[linecolor=blue](+... ...color=blue](2.6,0)(-3,-2.3335) \rput(-5.5,-3){.} \rput(5.5,3){.} \end{pspicture}")