5.§ 7 simplification and classification of Quadratic Curve Equations

§ 7 simplification and classification of Quadratic Curve Equations

ISimplification of Equations:

1Simplification of the central curve equation:

For the center curve F (x, y) = 0, let O '(,) as its center. If the coordinate origin is translated to o', the new equation will not include one term, then select the appropriate θ angle for rotation transformation, and eliminate the cross product items in the equation. Finally, the equation of the central curve can be reduced

(1)

Because the limit value is not 0, the center curve (1) is about the x' of the new system ′,

Y 'axis symmetry, that is, when the second main diameter of the central curve is used as the coordinate axis to create a new coordinate system, the equation of the curve is simplified to (1)

Example 1: Reduced quadratic curve equation x ²-xy + y ² + 4x-2y = 0

Solution:X + Y + 2 = 0, x-y + 2 = 0

Coordinate Transformation Formula

That is

X ''2 + 3y''-8 = 0

That is

2. Simplification of unintentional curve equations:

For the unintentional curve F (x, y) = 0, select the appropriate θ angle for Rotation Transformation to eliminate the cross product items in the equation, that is

The equation is simplified

Since there is only one shard and only one is 0, set it to 0, and then try again

Translation

Then the equation is simplified

(2)

Because

Thus unintentional curve (2) about x "axis symmetry, that is, the X" axis is its main diameter, and the intersection of the X "state and the curve is the coordinate origin of the new coordinate system.

It can be seen that the main diameter of the unintentional curve is used as the X 'axis, and the line perpendicular to the main diameter is used as the new line of the Y' axis, then the curve equation is simplified to (2)

Example 2: Reduced quadratic curve equation x 2 + 2xy + Y 2 + 2x-2y = 0

Solution:The x + y =-0 diameter of a given curve is x + y =-0, and the vertex of the curve is the origin. The x-y = 0 straight line is obtained and perpendicular to the main diameter, and take the coordinate transformation

That is

Substitute the original equation and simplify it

3. Simplification of the line-center curve equation:

For the line center curve F (x, y) = 0, take a center (,), and perform a translation transformation to remove one of the items in the equation, and then select the appropriate Alpha angle for Rotation Transformation, cross product items can also be eliminated, and the final equation can be simplified

Because the line has only one and only one is 0, you may wish to set the line center curve.

Simplified to (3)

Because the coordinate curve (3) is about the x-axis symmetry, it can be seen that the x-axis of the new coordinate system is the main diameter, that is, the first main diameter of the curve is used as the x-axis to create a new coordinate system, in the new system, the curve equation is simplified to (3)

Example 3:Reduced quadratic curve equation x 2-2xy + y 2 + 2x-2y = 0

Solution:It can be verified that the given curve is a line center curve, and its main diameter is x-y + 1 = 0. Then, take a straight line x + y = 0 perpendicular to the main diameter for coordinate transformation.

That is

Substituted into the original equation and simplified it

To sum up the method for simplifying the quadratic curve equation, we can draw the following conclusions:

Select the appropriate coordinate system to enable

The equation of the center quadratic curve is simplified

The equation of unintentional quadratic curve is simplified

The equation of the line center quadratic curve is simplified

Binary Quadratic Curve classification:

1 ° for the center curve, the equation can be reduced to (I)

When

A =, B =, then (I) is Ax 2 + By 2 = 1

If A, B> 0, make A =, B =, then (I) is

[1] -- elliptic

If A, B <0, make A =-, B =-, then (I) is

[2] ---- virtual elliptic

If A> 0, B <0, make A =, B =-, then (I) is

[3] ---- hyperbolic

Similarly, when A <0, B> 0, it is also A hyperbolic

When A =, B =, then (I) is

[4] ---- one point

Similarly, if a, B <0, then (I) is also a bit

If a> 0, B <0, make a =, B =-, then (I) is

[5] ----- two-Intersection Line

Similarly, if A <0, B> 0, then (I) is also a two-intersection line.

2 ° for unintentional curves, the equation can be reduced to (ii ),

P =, then (ii) is

[6] y² = 2px ------ parabolic

For a line-center curve, the equation can be reduced to (III ),

K =, then (iii) is y² = K

If K> 0, (iii) is

[7] y ² = A ² -------- two parallel lines,

If K is <0, (iii) is

[8] y ² =-A ² -------- two parallel lines,

If K = 0, (iii) is

[9] y² = 0 -------- dual straight line.