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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 ...
Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of circular sector area u and hyperbolic functions depending on hyperbolic sector area u. The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle.
Hyperbola: the midpoints of parallel chords lie on a line. Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes. The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram). The points of any chord may lie on different branches of the hyperbola.
An ellipse, its minimum bounding box, and its director circle. In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.
Like other rectangular hyperbolas, the orthocenter of any three points on the curve lies on the hyperbola. So, the orthocenter of the triangle A B C {\displaystyle ABC} lies on the curve. The line O I {\displaystyle OI} is tangent to this hyperbola at I {\displaystyle I} .
This line meets the circumcircle of ABC in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola. The general inconic is tangent to the three sidelines of ABC and is given by the equation + + =