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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.



Given two foci, the hyperbola can be defined as the locus of points, the difference of whose distance to the foci is constant.



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...



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...



We see the coordinates of the point at parametric location t on the hyperbola.



For the Hyperbola, we see that each axis intersection depends on only one parameter of the curve, the xaxis on a, the yaxis on b:



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 ...



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...



We contruct conjugate radii and project them onto the hyperbola axis. The areas formed by the two triangles are the same.



If a rectangular hyperbola circumscribes a triangle, it will also pass through the orthocenter of the triangle. A rectangular hyperbola has asymptotes...



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.



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.



If a rectangular hyperbola circumscribes a triangle, the center of the hyperbola lies on the 9 point circle of the triangle.



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 ...



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, ...



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...



Again we use the evolute, the envelope of the normals.



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...



