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A line through the focus and a line perpendicular to the directrix make the same angle with the tangent.



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...



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).



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.



The slope of the tangent at parametric location t gives a further characterization:



This is the directrix.



Take two nonparallel tangents of a parabola, and the angle at their intersection can be calculated.



We look at the caustic of the parabola y=x^2 formed by a set of rays at angle θ to the xaxis.



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 xaxis. We first creat...



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...



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 ...



If CD is a chord of a parabola and E its intersection with the directrix, then BE externally bisects the angle CBD.



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...



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 ...



The area of the triangle formed by 3 tangents is half the area of the triangle formed by their points of contact.



We use the evolute.



The locus of the intersection of tangents of given angles can be seen from the equation to be a hyperbola.



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.



We see that the subnormal is constant (half the parameter).



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...



We see that the ycoordinate is constant, and is in fact the ycoordinate of the directrix. Hence the orthocenter lies on the directrix.



We see that the line joining the intersection to the focus bisects the angle which the tangents’ points of contact make with the focus.



We see that the focus is the midpoint of the segment joining the intersections of the tangent and the normal with the axis:



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...



We look at the locus of the projection of the focus onto the normal, which turns about to be another parabola.



We find the coordinates of the intersection of the normals to the parabola.



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



Set the short sides along the axes with lengths a and ka. The equation is clearly a conic, and in fact a parabola:



Here is the evolute, expressed in terms of the parameter (the parameter p, being four times the distance between vertex and focus).



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 ...



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.



Here is the evolute, expressed in terms of the parameter (the parameter p, being four times the distance between vertex and focus).



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...



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...



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^(...



We find the intersection of two tangents to a parabola.



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...



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.



