Algebraic definition of conic

The Conics

A conic is the intersection of a plane and a right circular cone. The four basic types of conics are parabolas, ellipses, circles, and hyperbolas. We've already discussed parabolas and circles in previous sections, but here we'll define them a new way.

In the conics above, the plane does not pass through the vertex of the cone. When the plane does intersect the vertex of the cone, the resulting conic is called a degenerate conic. Degenerate conics include a point, a line, and two intersecting lines.

The equation of every conic can be written in the following form: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 . This is the algebraic definition of a conic. Conics can be classified according to the coefficients of this equation.

Circumference

A circumference or circle is the collection of points equidistant from a fixed point. The fixed point is called the center. The distance from the center to any point on the circle is the radius of the circle, and a segment containing the center whose endpoints are both on the circle is a diameter of the circle. The radius, r , equals one-half the diameter, d .

The standard equation for a circle is (x - h)2 + (y - k)2 = r 2 . The center is at (h, k) . The radius is r .

In a way, a circle is a special case of an ellipse. Consider an ellipse whose foci are both located at its center. Then the center of the ellipse is the center of the circle, a = b = r , and e = = 0 .

In prehistoric times, with the invention of the wheel is beginning to yield all the technology today, all thanks to this invention, the wheel, and even indirectly, and in this case we have applications girth.

In other aspects of life in which the presence of common use of the circles is in transport: mainly on the wheels and a clear example is the bicycle, a set of metal tubes with two wheels that apply geometry perfectly: the wheels are made of an"arc."

Circumference is a geometric factor of great importance. This very day everywhere, with this you can perform many techniques with high precision products such as CDs, watches, etc..

Parabola

As we saw in Quadratic Functions , a parabola is the graph of a quadratic function. As part of our study of conics, we'll give it a new definition. A parabola is the set of all points equidistant from a line and a fixed point not on the line. The line is called the directrix, and the point is called the focus. The point on the parabola halfway between the focus and the directrix is the vertex. The line containing the focus and the vertex is the axis. A parabola is symmetric with respect to its axis. Below is a drawing of a parabola.

If a parabola has a vertical axis, the standard form of the equation of the parabola is this: (x - h)2 = 4p(y - k) , where p≠ 0 . The vertex of this parabola is at (h, k) . The focus is at (h, k + p) . The directrix is the line y = k - p . The axis is the line x = h . If p> 0 , the parabola opens upward, and if p < 0 , the parabola opens downward.

If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y - k)2 = 4p(x - h) , where p≠ 0 . The vertex of this parabola is at(h, k) . The focus is at (h + p, k) . The directrix is the line x = h - p . The axis is the line y = k . If p > 0 , the parabola opens to the right, and if p < 0 , the parabola opens to the left. Note that this graph is not a function.

The parallel wave receiving antenna reflected on it and end up in the spotlight. Thanks to that we can watch TV for example.

Something very interesting that applies very much in China and helps save gas, money and effort are the parabolic solar cookers.

They consist of a parabolic screen that has reflective material which is supported by a metal frame that also holds a container holder.

The Ellipse

An ellipse

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