The earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolae. (The solution, however, does not meet the requirements imposed by compass and straightedge construction .) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes via the method of exhaustion in the third century BC, in his The Quadrature of the Parabola
The name "parabola" is due to Apollonius , who discovered many properties of conic sections. The focus–directrix property of the parabola and other conics is due to Pappus .
Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.
The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope . Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes , Marin Mersenne , and James Gregory . When Isaac Newton built the first reflecting telescope in 1668 he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror . Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers.
Let the directrix be the line x = − p and let the focus be the point ( p , 0). If ( x , y ) is a point on the parabola then, by Pappus' definition of a parabola, it is the same distance from the directrix as the focus; in other words:
X+P= √ (X-P)2+Y2
Squaring both sides and simplifying produces
as the equation of the parabola.
By translation, the above is the general equation of a parabola with a horizontal axis; by interchanging the roles of x and y one obtains the corresponding equation of a parabola with a vertical axis
so the graph of any function which is a polynomial of degree 2 in x is aparabola with a vertical axis.
More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation — one that does not factor as a product of two not necessarily distinct linear equations — of the general conic form
with the parabola restriction that
where all of the coefficients are real and where A and C are not both zero. The equation is irreducible if and only if the determinant of the 3×3 matrix
is non-zero: that is, if ( AC - B 2 /4) F + BED /4 - CD 2 /4 - AE 2 /4 ≠ 0. The reducible case, also called the degenerate case, gives a pair of parallel lines, possibly real, possibly imaginary, and possibly coinciding with each other.
A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolae are similar , meaning that while they can be different sizes, they are all the same shape. A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at infinity . The parabola is an inverse transform of a cardioid .
A parabola has a single axis of reflective symmetry , which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution.
The parabola is found in numerous situations in the physical world (see below).
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Latus rectum, semi-latus rectum, and polar coordinates
In polar coordinates , a parabola with the focus at the origin and the directrix parallel to the y -axis, is given by the equation.
r(1+cos0)=l
where l is the semilatus rectum : the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance from the focus to the vertex of the parabola or the perpendicular distance from the focus to the latus rectum.
The latus rectum is the chord that passes through the focus and is perpendicular to the axis. It has a length of 2l.
To derive the focus of a simple parabola, where the axis of symmetry is parallel to the y -axis with the vertex at (0,0), such as
then there is a point (0, f )—the focus, F —such that any point P on the parabola will be equidistant from both the focus and the linear directrix, L . The linear directrix is a line perpendicular to the axis of symmetry of the parabola (in this case parallel to the x axis) and passes through the point (0,- f ). So any point P=(x,y) on the parabola will be equidistant both to (0, f ) and ( x ,- f ).
FP , a line from the focus to a point on the parabola, has the same length as QP , a linedrawn from that point on the parabola perpendicular to the linear directrix, intersecting at point Q.
Imagine a right triangle with two legs, x and f-y (the vertical distance between F and P). The length of the hypotenuse, FP , is given by
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 2 f , and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle.
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A parabola is the collection of all points P in the plane that are the same distance from a fixed point F as they are from a fixed line D . The point F is called the focus of the parabola, and the line D is its directrix . As a result, a parabola is the set of points P for which d ( F , P ) = d ( P , D ) Let's sort out this definition by looking at a graph: focus directrix Take a line segment perpendicular to the directrix and intersect with a line segment from the focus of the same length. This will be a point on the parabola and will be the same distance from each. by symmetry we can get the other half 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
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Based on this definition and using the distance formula we can get a formula for the equation of a parabola with a vertex at the origin that opens left or right (see page 771 in book for derivation). a is the distance from the vertex to the focus (or opposite way for directrix) If the coefficient on x is positive the parabola opens to the right If the coefficient on x is negative the parabola opens to the left a a The equation for the parabola shown is: The parabola opens to the right and the vertex is 3 away from the focus. 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
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Let's find the focus and directrix of the parabola: This is 4 a Since the coefficient is negative, this parabola opens to the left. From the vertex count 4 in the negative direction to get the focus. focus (-4, 0) x = 4 a =4 a =4 The directrix is a line located the same distance from the vertex in the other direction.
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We could make a line segment through the focus of the parabola parallel to the directrix with endpoints on the parabola. This segment is called the latus rectum . focus (-4, 0) x = 4 a a latus rectum The length of the latus rectum is 4 a . 4 a This is very helpful information when graphing a parabola because we then know how wide the parabola is. The length of the latus rectum is 16 so it is 8 each way from the focus. (-4, 8) (-4, -8)
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In college algebra you graphed parabolas that opened up or down. The only difference with the equation is the x and the y are in different places. y = 1 Let's look at the steps to graph the parabola. (-2, -1) What direction does this open? If the x is squared it opens up or down (depending on the sign of the coefficient). If the y is squared it is right or left. What is a ? - 4 = - 4 a so a = 1. The focus is a away from the vertex in the direction the parabola opens. Draw the parabola containing these points. (0, -1) (2, -1) What is the length of the latus rectum? Add the directrix (not necessary for graphing but we want to see how it relates here). The directrix is a away from the vertex in the opposite direction as the focus. If the x is squared it opens up or down (depending on the sign of the coefficient). If the y is squared it is right or left. The length of the latus rectum is 4. Make a line segment 4 units long (2 each way) through the focus.
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Our parabola may have horizontal and/or vertical transformations. This would translate the vertex from the origin to some other place. The equations for these parabolas are the same but h is the horizontal shift and k the vertical shift: opens up The vertex will be at (h, k) opens down opens right opens left
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(-1, 2) Let's try one: Opens? y is squared and 8 is positive so right. Vertex? It is shifted to the left one and up 2 (set ( x + 1) = 0 and get x = -1 and set ( y - 2) = 0 and get y = 2). Vertex is (-1, 2) Focus? 4 a = 8 so a = 2. Focus is 2 away from vertex in direction parabola opens. Length of latus rectum? (1, 2) This is number in front of parenthesis which is 8, so 4 each way from focus. (1, 6) (1, -2) Directrix? "a" away from the vertex so 2 away in opposite direction of focus. x = -3
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(-2, 2) The secret to doing these is NOT to memorize a bunch of formulas given in your book in Tables 1 and 2, but to DRAW A PICTURE. What if we were given the focus of a parabola was (-2, 2) and the vertex was (-5, 2). If we draw a picture we can figure out the equation and anything else we need to know. (-5, 2) Just looking at this much graphed can you determine which way the parabola opens (and therefore what the standard form of the equation looks like) The focus must be inside the parabola so it must open to the right. Focus is " a " away from vertex so a = 3 3 simplified:
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The equation we are given may not be in standard form and we'll have to do some algebraic manipulation to get it that way. (you did this with circles in college algebra). Since y is squared, we'll complete the square on the y 's and get the x term to other side. middle coefficient divided by 2 and squared 1 1 must add to this side too to keep equation = factor Now we have it in standard form we can find the vertex, focus, directrix and graph.
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(-3/4, -1) Opens? Right ( y squared & no negative) Vertex? opposites of these values (-1, -1) Focus? 4 a = 1 so a = 1/4 Length of latus rectum? 1 1, so 1/2 each way Since the focus was at (-3/4, 1), to get the ends of the latus rectum, we'd need to increase the y value of the focus by 1/2 and then decrease the y value by 1/2. (look at the picture to determine this). (-3/4, -1/2) (-3/4, -3/2) Directrix? 1/4 away from vertex x = -5/4
A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance , causes the curve swept out to deviate slightly from the expected perfect parabola.
In nature, approximations of parabolae and paraboloids (such as catenary curves ) are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction ).
The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. He also later proved this
mathematically in his book Dialogue Concerning Two New Sciences . For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.
Another situation in which parabolae may arise in nature is in two-body orbits , for example, of a small planetoid or other object under the influence of the gravitation of the sun. Such parabolic orbits are a special case that are rarely found in nature. Orbits that form a hyperbola or an ellipse are much more common. In fact, the parabolic orbit is the borderline case between those two types of orbit. An object following a parabolic orbit moves at the exact escape velocity of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.
Approximations of parabolae are also found in the shape of the main cables on a simple suspension bridge . The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary , but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used. [9] [10] Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise hyperbolic cable is deformed toward a parabola. Unlike an inelastic chain, a freely-hanging spring of zero unstressed length takes the shape of a parabola
Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope .
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.
Vertical curves in roads are usually parabolic by design.
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There are many applications that involve parabolas. One is paraboloids of revolution. This is taking a parabola and revolving it to form "a dish". The waves come in and hit the surface and are reflected to the focus of the parabola. The receiver is placed at the focus. To work these problems, draw a picture with the vertex at the origin.
In algebraic geometry , the parabola is generalized by the rational normal curves , which have coordinates the standard parabola is the case n = 2, and the case n = 3 is known as the twisted cubic . A further generalization is given by the Veronese variety , when there are more than one input variable.
In the theory of quadratic forms , the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2 . Generalizations to more variables yield further such objects.
The curves y = x p for other values of p are traditionally referred to as the higher parabolas , and were originally treated implicitly, in the form x p = ky q for p and q both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula y = x p / q for a positive fractional power of x. Negative fractional powers correspond to the implicit equation x p y q = k , and are traditionally referred to as higher hyperbolas. Analytically, x can also be raised to an irrational power (for positive values of x ); the analytic properties are analogous to when x is raised to rational powers, but the resulting curve is no longer algebraic, and cannot be analyzed via algebraic geometry.
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