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A line through the focus and a line perpendicular to the directrix make the same angle with the tangent.
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Given a point (the focus) and a line (the directrix), a parabola can be defined as the locus of the points whose distance from the focus is equal to t...
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We take the parabola, whose vertex is (0,0) and whose focus is (0,a). The point at parametric location t is (2*a*t,a*t^2).
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Take two tangents of a parabola that are perpendicular to each other. The locus of all such points will be a line. This line is the directrix.
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The slope of the tangent at parametric location t gives a further characterization:
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This is the directrix.
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Take two nonparallel tangents of a parabola, and the angle at their intersection can be calculated.
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We look at the caustic of the parabola y=x^2 formed by a set of rays at angle θ to the x-axis.
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We wish to place the parabola 9x^2+16y^2+24xy+22x+46y+9=0 such that its vertex is at the origin an its focus is on the positive x-axis. We first creat...
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Let B be the focus of the parabola, and D be the intersection between a tangent whose point of contact is C and the parabola axis. The triangle BDC i...
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If we take the parabola Y=aX2 and look at the length of the horizontal chord through the focus, we find this is 1/a. Salmon calls this the parameter ...
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If CD is a chord of a parabola and E its intersection with the directrix, then BE externally bisects the angle CBD.
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Draw a tangent to a parabola and a line from the focus to that tangent. Draw another line from the tangency point to the vertex, and keep the angle be...
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We examine the locus of the intersection of the perpendicular from the focus to the tangent with the horizontal from the point of contact and find it ...
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The area of the triangle formed by 3 tangents is half the area of the triangle formed by their points of contact.
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We use the evolute.
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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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We look at the locus of the projection of the focus onto the normal, which turns about to be another parabola.
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We find the coordinates of the intersection of the normals to the parabola.
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We set the fixed point at the origin and the given line to be Y=b. The locus of the center of the circle is a parabola
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Set the short sides along the axes with lengths a and k-a. The equation is clearly a conic, and in fact a parabola:
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Here is the evolute, expressed in terms of the parameter (the parameter p, being four times the distance between vertex and focus).
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A is at the origin, B on the line X=a. We take the envelope of the line through B at angle η to AB. The equation is complicated, but clearly a ...
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The focal chord of curvature is the chord of the circle of curvature which passes through the focus. Its length is simply expressed in terms of t.
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Here is the evolute, expressed in terms of the parameter (the parameter p, being four times the distance between vertex and focus).
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The caustic curve (or catacaustic) is the envelope of a family of reflected rays. If a set of rays parallel to the axis of a parabola is reflected in...
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We take the family of parabolas with foci (0,0), whose generic member we pass through the point (0,-b). We look at the locus of the points of interse...
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If m is a quarter of the parameter of a parabola, and u and v are the lengths of two perpendicular tangents, we are to show that u^(2/3)/v^(4/3) + v^(...
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We find the intersection of two tangents to a parabola.
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We take the points at x locations x_0, x_0+h/2, and x_0+h on the curve Y=X^2, and we create triangles by connecting the points and their tangents. Not...
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If we create a parametric curve where both x and y are quadratic in t, and inspect its implicit equation, we see that it is a conic.
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