# symmetry of parabola

Let three tangents to a parabola form a triangle. The blue orbit is the Earth's. −

= P

P

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's "Vomit Comet", follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.

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V π

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Remark 2: The second polar form is a special case of a pencil of conics with focus

t

v 0 Since triangles △FBE and △CBE are congruent, FB is perpendicular to the tangent BE.

Application: The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point , one gets the implicit representation. Q {\displaystyle m_{0}} In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. x Every parabola has an axis of symmetry and, as the graph shows, the graph to either side of the axis of symmetry is a mirror image of the other side. |

2 Draw perpendicular ST intersecting BQ, extended if necessary, at T. At B draw the perpendicular BJ, intersecting VX at J. yields. = i x

a {\displaystyle P_{0}=(x_{0},y_{0}),P_{1}=(x_{1},y_{1}),P_{2}=(x_{2},y_{2})}

0 ,

The curves y = xp for other values of p are traditionally referred to as the higher parabolas and were originally treated implicitly, in the form xp = kyq for p and q both positive integers, in which form they are seen to be algebraic curves. x As in all cases in the physical world, the trajectory is always an approximation of a parabola.

}, Analogous to the inscribed angle theorem for circles, one has the inscribed angle theorem for parabolas:[11][12], (Proof: straightforward calculation: If the points are on a parabola, one may translate the coordinates for having the equation

) From the above, the area of the parabolic sector

x

a 2

of the perpendicular from the focus , [1] The focus–directrix property of the parabola and other conic sections is due to Pappus. The right side of the diagram shows part of this parabola. Parabolic microphone with optically transparent plastic reflector, used to overhear referee conversations at an American college football game. )

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then (1) the distance from F to T is 2f, and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle.[13]:p.26.

1 2

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= Physicist Stephen Hawking in an aircraft flying a parabolic trajectory to simulate zero gravity, Intersection of a tangent and perpendicular from focus, Reflection of light striking the convex side, Two tangent properties related to the latus rectum, Focal length calculated from parameters of a chord, Area enclosed between a parabola and a chord, Corollary concerning midpoints and endpoints of chords, A geometrical construction to find a sector area, Focal length and radius of curvature at the vertex. x A onto the directrix

and

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