Parabolas

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The locus of the intersection of tangents of given angles can be seen from the equation to be a hyperbola.
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Take two tangents and their respective normals. Set the parameters of the tangency points to be t and -1/t, and the normals are thus perpendicular.
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We see that the subnormal is constant (half the parameter).
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The orthocenter of the triangle formed by three tangents to a parabola lies on the directrix. We create the parabola with vertex at the origin and foc...
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We see that the y-coordinate is constant, and is in fact the y-coordinate of the directrix. Hence the orthocenter lies on the directrix.
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We see that the line joining the intersection to the focus bisects the angle which the tangents’ points of contact make with the focus.
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We see that the focus is the midpoint of the segment joining the intersections of the tangent and the normal with the axis:
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Given a parabola with vertex A and focus B and a chord CD let E be the intersection of the tangents at C and D. We create the line perpendicular to t...
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