5 § 1 Introduction of complex elements, intersection of quadratic curves and straight lines

§ 1 Introduction of complex elements, intersection of quadratic curves and straight lines

IComplex elements on the plane

A Cartesian coordinate system {O; I, j} is created on the plane. The concept of vertices on the plane is expanded as follows: any pair of ordered plural numbers (x, y) it is the coordinate of a point p on the plane. If x and y are all real numbers, p is called the real point. Otherwise, p is called the virtual point, and the real point and the virtual point are collectively referred to as the complex point. After the concept of point is expanded, the original real plane becomes a complex plane.

The following elements are introduced in the complex plane:

(1) Complex Vector: the complex vector with (,) as the starting point and (,) as the end point is defined:

= (-) I + (-) j, where-,-is called its component and is recorded

{-,-}, The vector with different components is called a virtual vector; otherwise, it is called a real vector. If the corresponding component of the two vectors is proportional, the two vectors are called parallel.

(2) complex line: In the Cartesian system, one equation

Ax + by + c = 0 (a, B is the plural)

The displayed graph is called a complex straight line. If a, B, and c correspond to three real numbers, it is called a real straight line; otherwise, it is called a virtual straight line. Note: real straight lines can have virtual points.

Note: There are infinite complex points on the real line, but there is only one real point on the virtual line.

(3) set this shard:

With (,), (,), if the coordinate M (x, y) is full

X =, y = (λ =-1)

M is the linear point of the line segment, and λ is the ratio. In particular, the midpoint is ()

(4) complex elements:

If the sum is a pair of the pair, P (,) and (,) are called a pair of bounded complex points. It is obvious that the real point is bounded with its own; the midpoint of the link segment of the two centers of the complex points is the real point. If two straight lines

: X + y + = 0, I = 1, 2 is satisfied with the constructor, And the constructor, And the constructor, is called a pair of the constructor complex line. If the corresponding component of the two vectors is the constructor complex, this is called the two vectors as the bounded complex vectors.

Note: The distance formula on the real plane cannot be generalized on the complex plane. This is because, on the real plane

D ² = (-) ² + (-) ²

On the complex plane, the upper right side is a complex number, and its square root has two equal values, which cannot determine the distance between two points.

IIIntersection of the quadratic curve and the (real) line:

Has a quadratic curve C: F (x, y) ≡ x² + 2xy + y ² + 2x + 2y + = 0 and straight line

L:

In order to study the intersection of l and c, we also need to study the t value that satisfies the following equation:

(X² + 2XY + y²) t² + 2 {(++) X + (++) Y}

+ F (,) = 0

That is, Phi (X, Y) t² + 2 [(,) X + (,) Y] t + F (,) = 0 (*)

(1) If PHI (X, Y) is less than 0

(*) Discriminant

△= 4 {[(,) X + (,) Y] ²-phi (X, Y) F (,)}

When △> 0, l and C are handed in two different real points;

When △< 0, l and C are handed over to two different virtual points;

When △= 0, l and C are handed over to the dual joint real point -- tangent

(Ii) If PHI (X, Y) = 0

When (,) X + (,) Y =0, (*) has a solid root, so that C and l are only handed over to one real point (not tangent)

When (,) X + (,) Y = 0, but F (,) =0, (*) is the contradiction equation.

C and l are not allowed

When (,) X + (,) Y = F (,) = 0, (*) is an equality.

∴ L on quadratic curve C -- tangent