Notes : CONIC SECTIONS

CONIC SECTIONS

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. Study the figures below to see how a conic is geometrically defined.

Figure 1 : The four basic types of conics

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.

The determinant of the equation is B 2 - 4AC . Assuming a conic is not degenerate, the following conditions hold true: If B 2 -4AC > 0 , the conic is a hyperbola. If B 2 -4AC < 0 , the conic is a circle, or an ellipse. If B 2 - 4AC = 0 , the conic is a parabola.

Another way to classify conics has to do with the product of A and C . Assuming a conic is not degenerate, the following conditions hold true: if AC> 0 , the conic is an ellipse or a circle. If AC < 0 , the conic is a hyperbola. If AC = 0 , and A and C are not both zero, the conic is a parabola. Finally, if A = C , the conic is a circle.

In the following sections we'll study the other forms in which the equations for certain conics can be written, and what each part of the equation means graphically.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: -3x 2 + y + 2 = 0 ?

It is a parabola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: 2x 2 +3xy - 4y 2 + 2x - 3y + 1 = 0 ?

It is a hyperbola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: 2x 2 -3y 2 = 0 ?

It is a hyperbola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: -3x 2 + xy - 2y 2 + 4 = 0?

It is an ellipse or a circle.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: x = 0 ?

It is a degenerate conic. x = 0 is a line.

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.

Figure %: In the parabola above, the distance d from the focus to a point on the parabola is the same as the distanced from that point to the directrix.

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.

Let P = (x, y) be a point on a parabola. Let l be the tangent line to the parabola at the point P . Let  be a line segment whose endpoints are the focus of the parabola and P . Every parabola has the following property: the angle θ between the tangent line l and the segment  equal to the angle μ between the tangent line and the axis of the parabola. This means (in a physical interpretation) that a beam sent from the focus to any point on the parabola is reflected in a line parallel to the axis. Furthermore, if a beam traveling in a line parallel to the axis contacts the parabola, it will reflect to the focus. This is the principle on which satellite dishes are built.

Figure 2 : The reflective property of a parabola: θ = μ

ELLIPSE AND CIRCLE

An ellipse is the set of points such that the sum of the distances from any point on the ellipse to two other fixed points is constant. The two fixed points are called the foci (plural of focus) of the ellipse.

Figure 3 : The sum of the distances d 1 + d 2 is the same for any point on the ellipse.

The line segment containing the foci of an ellipse with both endpoints on the ellipse is called the major axis. The endpoints of the major axis are called the vertices. The point halfway between the foci is the center of the ellipse. The line segment perpendicular to the major axis and passing through the center, with both endpoints on the ellipse, is the minor axis.

The standard equation of an ellipse with a horizontal major axis is the following:  +  = 1 . The center is at (h, k) . The length of the major axis is 2a , and the length of the minor axis is 2b . The distance between the center and either focus is c , where c 2 =a 2 - b 2 . Here a > b > 0 .

The standard equation of an ellipse with a vertical major axis is the following:  +  = 1 . The center is at (h, k) . The length of the major axis is 2a , and the length of the minor axis is 2b . The distance between the center and either focus is c , where c 2 =a 2 - b 2 . Here a > b > 0 .

The eccentricity of an ellipse is e =  . For any ellipse,0 < e < 1 . The eccentricity of an ellipse is basically a measure of the "ovalness" of an ellipse. It is the ratio of the distance between the foci and the length of the major axis. If the foci are very near the center of an ellipse, the ellipse is nearly circular, and e is close to zero. If the foci are relatively far away from the center, the ellipse is shaped more like an oval, and e is closer to one.

Circles

A 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 .

HYPERBOLA

A hyperbola is the set of all points such that the difference of the distances between any point on the hyperbola and two fixed points is constant. The two fixed points are called the foci of the hyperbola.

Figure 4 : The difference of the distances d 1 - d 2 is the same for any point on the hyperbola.

The graph of a hyperbola is not continuous--every hyperbola has two distinct branches. The line segment containing both foci of a hyperbola whose endpoints are both on the hyperbola is called the transverse axis. The endpoints of the transverse axis are called the vertices of the hyperbola. The point halfway between the foci (the midpoint of the transverse axis) is the center.

The standard equation for a hyperbola with a horizontal transverse axis is  -  = 1 . The center is at (h, k) . The distance between the vertices is 2a . The distance between the foci is 2c . c 2 = a 2 +b 2 . The line segment of length 2b perpendicular to the transverse axis whose midpoint is the center is the conjugate axis of the hyperbola.

The standard equation for a hyperbola with a vertical transverse axis is  -  = 1 . The center is at (h, k) . The distance between the vertices is 2a . The distance between the foci is 2c . c 2 = a 2 + b 2 .

Every hyperbola has two asymptotes. A hyperbola with a horizontal transverse axis and center at (h, k)has one asymptote with equation y = k + (x - h) and the other with equation y = k - (x - h) . A hyperbola with a vertical transverse axis and center at (h, k) has one asymptote with equation y = k + (x - h) and the other with equation y = k - (x - h) .

The eccentricity of a hyperbola, like an ellipse, is e =  . For all hyperbolas, though, c > a , so e > 1 . If e is close to one, the branches of the hyperbola are very narrow, but if e is much greater than one, then the branches of the hyperbola are very flat.

~ Destress Corner ~