Hyperbolas

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A ray of light projected from one focus and reflected in a normal to the hyperbola will reflect to the other focus.
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Given two foci, the hyperbola can be defined as the locus of points, the difference of whose distance to the foci is constant.
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Given a point (the focus) and a line (the directrix), we examine the locus of the points whose distance from the focus is k times the distance from th...
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We see that if we have the hyperbola and the conjugate hyperbola, then the points at parameter locations t on one hyperbola and t on the other lie on...
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We see the coordinates of the point at parametric location t on the hyperbola.
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For the Hyperbola, we see that each axis intersection depends on only one parameter of the curve, the x-axis on a, the y-axis on b:
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We look at the locus of the intersections of the tangents at the end of the diameter and the focal chord through a point. This locus is a circle with ...
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The equivalent to defining an ellipse with the equation (x^2/a^2)+(y^2/b^2)=1 , is to define a hyperbola with equation (x^2/a^2)-(y^2/b^2)=1 . We al...
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We contruct conjugate radii and project them onto the hyperbola axis. The areas formed by the two triangles are the same.
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If a rectangular hyperbola circumscribes a triangle, it will also pass through the orthocenter of the triangle. A rectangular hyperbola has asymptotes...
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If two lines have the equations A(x,y)=0 and B(x,y)=0, then the conic AB=a is a hyperbola with A and B as asymptotes.
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The area of the triangle whose vertices are the center of the hyperbola and the intersections of a tangent line with the asymptotes is constant.
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If a rectangular hyperbola circumscribes a triangle, the center of the hyperbola lies on the 9 point circle of the triangle.
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We take a point and a constrained subtended angle on a horizontal line. The locus of the center of the circumcircle of the resulting triangle forms a ...
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For a given ellipse, we take the tangents through a fixed point. We take the midpoint of the segment of a general tangent cut off by these tangents, ...
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To generate area k, we use one short side length a, the other length 2k/a. Clearly the locus is a hyperbola. We can find its foci by drawing a hyperb...
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Again we use the evolute, the envelope of the normals.
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Take 2 circles of the same radius in 3 dimensions, such that the circles lie in parallel planes and the line joining their centers is perpendicular to...
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