What conic section is generated when the plane is parallel to one generator and intersecting only one of its nappes?

A conic section is the intersection of a plane and a double right circular cone . By changing the angle and location of the intersection, we can produce different types of conics. There are four basic types: circles , ellipses , hyperbolas and parabolas . None of the intersections will pass through the vertices of the cone.

If the right circular cone is cut by a plane perpendicular to the axis of the cone, the intersection is a circle. If the plane intersects one of the pieces of the cone and its axis but is not perpendicular to the axis, the intersection will be an ellipse. To generate a parabola, the intersecting plane must be parallel to one side of the cone and it should intersect one piece of the double cone. And finally, to generate a hyperbola the plane intersects both pieces of the cone. For this, the slope of the intersecting plane should be greater than that of the cone.

The general equation for any conic section is

A x 2 + B x y + C y 2 + D x + E y + F = 0 where A , B , C , D , E and F are constants.

As we change the values of some of the constants, the shape of the corresponding conic will also change. It is important to know the differences in the equations to help quickly identify the type of conic that is represented by a given equation.
      If B 2 − 4 A C is less than zero, if a conic exists, it will be either a circle or an ellipse.
      If B 2 − 4 A C equals zero, if a conic exists, it will be a parabola.
      If B 2 − 4 A C is greater than zero, if a conic exists, it will be a hyperbola.

STANDARD FORMS OF EQUATIONS OF CONIC SECTIONS:

Circle	( x − h ) 2 + ( y − k ) 2 = r 2	Center is ( h , k ) . Radius is r .
Ellipse with horizontal major axis	( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1	Center is ( h , k ) . Length of major axis is 2 a . Length of minor axis is 2 b . Distance between center and either focus is c with c 2 = a 2 − b 2 , a > b > 0 .
Ellipse with vertical major axis	( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1	Center is ( h , k ) . Length of major axis is 2 a . Length of minor axis is 2 b . Distance between center and either focus is c with c 2 = a 2 − b 2 , a > b > 0 .
Hyperbola with horizontal transverse axis	( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1	Center is ( h , k ) . Distance between the vertices is 2 a . Distance between the foci is 2 c . c 2 = a 2 + b 2
Hyperbola with vertical transverse axis	( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1	Center is ( h , k ) . Distance between the vertices is 2 a . Distance between the foci is 2 c . c 2 = a 2 + b 2
Parabola with horizontal axis	( y − k ) 2 = 4 p ( x − h ) , p ≠ 0	Vertex is ( h , k ) . Focus is ( h + p , k ) . Directrix is the line x = h − p Axis is the line y = k
Parabola with vertical axis	( x − h ) 2 = 4 p ( y − k ) , p ≠ 0	Vertex is ( h , k ) . Focus is ( h , k + p ) . Directrix is the line y = k − p . Axis is the line x = h

Solving Systems of Equations

You must be familiar with solving system of linear equation . Geometrically it gives the point(s) of intersection of two or more straight lines. In a similar way, the solutions of system of quadratic equations would give the points of intersection of two or more conics.

Algebraically a system of quadratic equations can be solved by elimination or substitution just as in the case of linear systems.

Example:

Solve the system of equations.

x 2 + 4 y 2 = 16 x 2 + y 2 = 9

The coefficient of x 2 is the same for both the equations. So, subtract the second equation from the first to eliminate the variable x . You get:

3 y 2 = 7

Solving for y :

3 y 2 3 = 7 3 y 2 = 7 3 y = ± 7 3

Use the value of y to evaluate x .

x 2 + 7 3 = 9 x 2 = 9 − 7 3 = 20 3 x = ± 20 3

Therefore, the solutions are ( + 20 3 , + 7 3 ) , ( + 20 3 , − 7 3 ) , ( − 20 3 , + 7 3 ) and ( − 20 3 , − 7 3 ) .

Now, let us look at it from a geometric point of view.

If you divide both sides of the first equation x 2 + 4 y 2 = 16 by 16 you get x 2 16 + y 2 4 = 1 . That is, it is an ellipse centered at origin with major axis 4 and minor axis 2 . The second equation is a circle centered at origin and has a radius 3 . The circle and the ellipse meet at four different points as shown.

Conic Sections Conics, a Family of Similarly Shaped Curves - Properties of Conics

Conics, a Family of Similarly Shaped Curves - Properties of Conics

Dandelin's Spheres - proof of conic sections focal properties Proof that conic section curve is the ellipse Proof that conic section curve is the hyperbola Proof that conic section curve is the parabola

Conics - a family of similarly shaped curves Conics, a family of similarly shaped curves - properties of conics

By intersecting either of the two right circular conical surfaces (nappes) with the plane perpendicular to the axis of the cone the resulting intersection is a circle c, as is shown in the figure.

When the cutting plane is inclined to the axis of the cone at a greater angle than that made by the generating segment or generator (the slanting edge of the cone), i.e., when the plane cuts all generators of a single cone, the resulting curve is the ellipse e.

Thus, the circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone.

If the cutting plane is parallel to any generator of one of the cones, then the intersection curve is the parabola p.

When the cutting plane is inclined to the axis at a smaller angle than the generator of the cone, i.e., if the intersecting plane cuts both cones the hyperbola h is generated.

Dandelin Spheres - proof of conic sections focal properties Proof that conic section curve is the ellipse In the case when the plane E intersects all generators of the cone, as in down figure, it is possible to inscribe two spheres which will touch the conical surface and the plane. Upper sphere touches the cone surface in a circle k1 and the plane at a point F1. Lower sphere touches the cone surface in a circle k2 and the plane at a point F2.

Arbitrary chosen generating line g intersects the
circle k1 at a point M, the circle k2 at a point N and
the intersection curve e at a point P.

We see that points, M and F1 are the tangency
points of the upper sphere and points, N and F2 the tangency points of the lower sphere of the tangents

drawn from the point

P exterior to the spheres.

Since the segments of tangents from a point exterior to sphere to the points of contact, are equal

PM = PF1 and PN = PF2.

And since planes of circles k1and k2, are parallel, then are all corresponding generating segments equal

MN = PM + PN is constant.

Thus, the intersection curve is the locus of points in the plane for which sum of distances from the two fixed points F1and F2, is constant, i.e., the curve is the ellipse.

The proof is due to the French/Belgian mathematician Germinal Dandelin (1794 – 1847). Proof that conic section curve is the hyperbola When the intersecting plane is inclined to the vertical axis at a smaller angle than does the generator of the

cone, the plane cuts both cones creating the hyperbola h which therefore consists of two disjoining branches as shows the right figure.

Inscribed spheres touch the plane on the same side at points F1 and F2 and the cone surface at circles k1and k2.

The generator g intersects the circles k1and k2, at points, M and N, and the intersection curve at the point P.

By rotating the generator g around the vertex V by
360°, the point P will move around and trace both
branches of the hyperbola.

While rotating, the generator will coincide with the plane two times and then will have common points with the curve only at infinity.

As the line segments, PF1and PM are the tangents segments drawn from P to the upper sphere, and the segments PF2 and PN are the tangents segments drawn to the lower sphere, then

PM = PF1 and PN = PF2.

Since the planes of circles k1 and k2, are parallel, then are all generating segments from k1to k2 of equal length, so

MN = PM - PN or PF1 - PF2 is constant.

Thus, the intersection curve is the locus of points in
the plane for which difference of distances from the two fixed points F1 and F2, is constant, i.e., the curve is the hyperbola.

Proof that conic section curve is the parabola When the cutting plane is parallel to any generator of one of the cones then we can insert only one sphere into the cone which will touch the plane at the point F and the cone surface at the circle k. Arbitrary chosen generating line g intersects the circle k at a point M, and the intersection curve p at a point P. The point P lies on the circle k' which is parallel with the plane K as shows the down figure. By rotating the generator g around the vertex V, the point P will move along the intersection curve.

While the generator approaches position to be parallel to the plane E, the point P will move far away from F. That shows the basic property of the parabola that the line at infinity is a tangent.

The segments, PF and PM belong to tangents drawn from P to the sphere

so, PM = PF.

Since planes of the circles, k and k' are parallel to each other and perpendicular to the section through the cone axis, and as the plane E is parallel to the slanting edge VB, then the intersection d, of planes E and K, is also perpendicular to the section through the cone axis.

Thus, the perpendicular PN from P to the line d,

PN = BA = PM or PF = PN.

Therefore, for any point P on the intersection curve the distance from the fixed point F is the same as it is from the fixed line d, it proves that the intersection curve is the parabola. Conics - a family of similarly shaped curves A conic is the set of points P in a plane whose distances from a fixed point F (the focus) and a fixed line d (the directrix), are in a constant ratio. This ratio named the eccentricity e determines the shape of the curve. We can see that conics represent a family of similarly shaped curves if we write their equations in vertex form. Recall the method we used to transform equations of the ellipse and the hyperbola from standard to vertex form. We placed the vertex of the curve at the origin translating its graph. Thus, obtained are their vertex equations;

y2 = 2px - (p/a)x2

- the ellipse and the circle

(for the circle p = a = r)

y2 = 2px + (p/a)x2

- the hyperbola

Using geometric interpretation of these equations we compare the area of the square y2, formed by the ordinate of a point P(x, y), with the area of the rectangle 2p · x, whose one side is the abscissa x of the point P and the parameter 2p other side, it follows that

- for the ellipse the area of the square is smaller, than the area of the rectangle, - for the parabola is equal, - for the hyperbola the area of the square is greater than the area of the rectangle. The names of curves were given as a result of the above relations, so; - the word “ellipse” (elleipyis) in Greek means “deficiency," - the word “parabola” (parabolh) means “equality” and - the word “hyperbola” (uperbolh) means “excess.” In the given vertex equations we can make following substitutions for;

- the ellipse

- the circle

p = a = b = r => e = 0

- the hyperbola

Thus, the equation of conics in vertex form is

y2 = 2px - (1 - e2)x2.

The values of e define the curve the conic section makes, such that for e = 0 - a circle, 0 < e < 1 - an ellipse, e = 1 - a parabola, e > 1 - a hyperbola, as shows the above figure.

Conic sections contents