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A consequence is that the equation (in ,) of the parabola determined by 3 points = (,), =,,, with different x coordinates is (if two x coordinates are equal, there is no parabola with directrix parallel to the x axis, which passes through the points) =.
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 family of conic sections of varying eccentricity share a focus point and directrix line, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally separated ...
The directrix has equation = . With = (,), the ... This is the equation of an ellipse (<), or a parabola (=), or a hyperbola (>). All of these non-degenerate conics ...
(1) All rulings are parallel to a plane, the directrix plane. (2) All rulings intersect a fixed line, the axis. The conoid is a right conoid if its axis is perpendicular to its directrix plane. Hence all rulings are perpendicular to the axis. Because of (1) any conoid is a Catalan surface and can be represented parametrically by
The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at (x 0, y 0) orthogonally, the following equations hold: = = The last equation is equivalent to =, which is the equation of the directrix.
The directrix is often taken as a plane curve, in a plane not containing the apex, but this is not a requirement. [1] In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve ...
In this position, the hyperbolic paraboloid opens downward along the x-axis and upward along the y-axis (that is, the parabola in the plane x = 0 opens upward and the parabola in the plane y = 0 opens downward). Any paraboloid (elliptic or hyperbolic) is a translation surface, as it can be generated by a moving parabola directed by a second ...