Analysis of a conic

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ANALYSIS OF A CONIC

The general equation of the second degree

1) f(x, y) = ax² + 2hxy + by² + 2gx + 2fy + c = 0

represents one of 9 different conics:

Any equation of the second degree can be reduced to one of the nine canonical forms by a suitable rotation and translation of the coordinate system. Thus when faced with a particular second degree equation the following questions immediately present themselves:

1] Which of the 9 conics does this equation represent?

2] What is the exact equation of the conic when in canonical form? That is, what is its equation when expressed with respect to the canonical coordinate system?

3] What is the location of the origin of the canonical system?

4] What is the orientation of the canonical coordinate system? In other words, in what directions do the canonical system axes point?

We will now deal with the procedure used in answering these questions. Our starting point is our given equation

2) f(x, y) = ax² + 2hxy + by² + 2gx + 2fy + c = 0

which represents some conic located somewhere in the plane. In Figure 1 a conic (an ellipse) is shown located at some point in the plane. Figure 1 also shows the original x-y coordinate system along with two other coordinate systems – an intermediate x'- y' coordinate system and the x_c- y_c canonical coordinate system. We wish to know the location and orientation of the canonical coordinate system and the exact equation of the conic as referred to that system. Figure 1 shows the x'- y' coordinate system as a system obtained by rotating the x-y system by θ degrees about its origin. θ represents the rotation required to eliminate the xy term in the original equation as computed from

A rotation of this amount will put its axes parallel to the axes of the canonical system.

We proceed as follows:

1. We obtain the equation of the conic as referred to the intermediate x'- y' coordinate system by substituting into equation 2) the expressions for x and y in terms of x' and y' as given by the rotational transformation equations

x = x' cos θ - y' sin θ

y = x' sin θ + y' cos θ

where the θ in these equations is that particular value of θ computed from 3) above.

On expanding and simplifying we then have the equation

4) g(x', y') = 0

as the equation of the conic as referred to the intermediate x'- y' coordinate system.

2. We determine translation required to carry the x'- y' system into the canonical x_c- y_c system. In other words, we determine the values of h' and k' where (h', k') is the origin of the canonical x_c- y_c system with respect to the x'- y' system. In the case of the central conics (ellipses and hyperbolas) this translation corresponds to that translation that will eliminate both the x and y terms from the equation.

3. We obtain the equation of the conic as referred to the canonical x_c- y_c coordinate system by substituting into equation 4) the expressions for x' and y' in terms of x_c and y_c as given by the translation transformation equations

x' = x_c + h'

y' = y_c + k' .

This process gives us the answers to our original questions. The mechanics of substituting into equations, expanding, and simplifying can be laborious and fortunately, in the case of the central conics, it can be bypassed. By utilizing certain invariant quantities and employing some abstract results from matrix theory we can answer the questions without going to all that labor.

Analysis procedure.

First we define the following quantities related to the equation

f(x, y) = ax² + 2hxy + by² + 2gx + 2fy + c = 0 :

λ₁, λ₂ – the characteristic roots of matrix e i.e. the roots of the characteristic equation

which can be written as

4) λ² - Iλ + J = 0

Conic identification. We identify the conic by use of the following table:

Case	∇	J	∇/ J	K	Conic
1	≠0	> 0	< 0		Ellipse
2	≠0	> 0	> 0		Imaginary ellipse
3	0	> 0			Pair of imaginary lines intersecting in a real point
4	≠0	< 0			Hyperbola
5	0	< 0			A pair of intersecting lines
6	≠0	0			Parabola
7	0	0		< 0	A pair of parallel lines
8	0	0		> 0	A pair of imaginary parallel lines
9	0	0		0	A pair of coincident straight lines

Analysis of central conics (ellipses and hyperbolas).

Orientation of the canonical coordinate system. The orientation of the canonical system is found by computing the rotation angle θ required to eliminate the xy term. The formula is

Location of the origin of the canonical system. The location of the origin (x₀, y₀) of the canonical system with respect to the x-y system is given by solving the following system of equations for x₀, y₀ :

ax₀ + hy₀ + g = 0

hx₀ + by₀ + f = 0

Equation of the conic in the canonical system. The equation of the conic with respect to the canonical system is

λ₁x² + λ₂y²+ c' = 0

where

c' = gx₀ + fy₀ + c

and λ₁, λ₂ are obtained by computing the roots of characteristic equation

λ² - Iλ + J = 0 ,

or more explicitly,

λ² - (a + b)λ + ab - h² = 0