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k = 1 is the tangent line to the right of the circles looking from c 1 to c 2. k = −1 is the tangent line to the right of the circles looking from c 2 to c 1. The above assumes each circle has positive radius. If r 1 is positive and r 2 negative then c 1 will lie to the left of each line and c 2 to the right, and the two tangent lines will ...
A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a secant line, at one point a tangent line and at no points an exterior line. A chord is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line ...
The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem.
The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x 1, y 1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x 1, y 1), so it has the form (x 1 − a)x + (y 1 – b)y = c.
The secants ′ ¯, ′ ¯ meet on the radical axis of the given two circles. Moving the lower secant (see diagram) towards the upper one, the red circle becomes a circle, that is tangent to both given circles. The center of the tangent circle is the intercept of the lines ¯, ¯. The secants ′ ¯, ′ ¯ become tangents at the points ,.
This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows: = + ′ ().
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Tweak the tangent so it is indeed tangent. 05:38, 4 December 2005: 612 × 618 (13 KB) Optimager~commonswiki: A circle showing the chord, secant, and tangent. This is a vector graphic version of Image:Circle lines.png, originally made by User:Jleedev on March 20, 2005 using Inkscape. The original was released under th
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