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For a parametric equation of a parabola in general position see § As the affine image of the unit parabola. The implicit equation of a parabola is defined by an irreducible polynomial of degree two: + + + + + =, such that =, or, equivalently, such that + + is the square of a linear polynomial.
Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled t ; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for ...
Define b by the equations c 2 = a 2 − b 2 for an ellipse and c 2 = a 2 + b 2 for a hyperbola. For a circle, c = 0 so a 2 = b 2, with radius r = a = b. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In standard form the parabola will always pass through the ...
A circular paraboloid contains circles. This is also true in the general case (see Circular section). From the point of view of projective geometry, an elliptic paraboloid is an ellipsoid that is tangent to the plane at infinity. Plane sections. The plane sections of an elliptic paraboloid can be: a parabola, if the plane is parallel to the axis,
This is the equation of a parabola, so the path is parabolic. The axis of the parabola is vertical. If the projectile's position (x,y) and launch angle (θ or α) are known, the initial velocity can be found solving for v 0 in the afore-mentioned parabolic equation:
For solving the cubic equation x 3 + m 2 x = n where n > 0, Omar Khayyám constructed the parabola y = x 2 /m, the circle that has as a diameter the line segment [0, n/m 2] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis.
To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r 1 and r 2. To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
The scale factors for the parabolic coordinates (,) are equal = = + Hence, the infinitesimal element of area is = (+) and the Laplacian equals = + (+) Other differential operators such as and can be expressed in the coordinates (,) by substituting the scale factors into the general formulae found in orthogonal coordinates.