5 § 3 tangent and singularity of quadratic curves

§ 3 tangent and singularity of quadratic curves

ITangent:

1,Definition: If a straight line L and the quadratic curve C are placed on a dual combination real point, or l is on the quadratic curve C, it is called L

For cTangent. The common point of tangent and C is calledCut Point.

2,Method:

Set (,) in C to the tangent l of the cut point:

X: Y

1 ° When (,), (,) is not all 0,

If X: Y is not the same direction, L is tangent to C. <strong> L and C are handed over to the dual solid point.

<Strong> △= [(,) x + (,) y] ²-phi (x, y) f (,) = 0

<Strong> (,) x + (,) y = 0 <strong> X: y = -(,):(,)

If X: Y is the progressively direction, L is tangent to C. <strong> L is in C. <strong> 〉

(,) X + (,) y = 0 <strong> X: y = -(,):(,)

Thus tangent L:

That is, (,) (X-) + (,) (Y-) = 0

(,) X + (,) Y-[(,) + (,)] = 0

(,) X + (,) Y + (,) = 0

That is

X + (Y + x) + Y + (x +) + (Y +) + = 0 (*)

Note: When (,) and (,) are not all 0, (*) is the tangent equation with (,) as the cut point. It is not hard to see that if (,) makes (,), (,) Not all 0, it is required to think of the tangent of the cut point, just in the equation of C

X, Y ,,

Replace x 2 xy Y 2 x y

You can.

2 ° When (,) = (,) = 0,

For a line over and along the non-gradual direction l :,

△= [(,) X + (,) Y] ²-phi (X, Y) F (,) = 0

∴ L is the tangent, and for any line over and along the gradual direction l:

Phi (X, Y) = (,) X + (,) Y = F (,) = 0,

∴ L is in the curve, that is, l is also a tangent.

It can be seen that if the curve is a little above (,) (, y .) = (,) = 0, then any line passing through is the tangent of C. In this case, the tangent of C is usually taken only from a straight line in the same direction.

 

ErqiPoint:

1,Definition: The point on the quadratic curve that the coordinates satisfy is calledSingularity. The non-Singularity on the quadratic curve is also calledNormal.

Visible:

1 ° (,) is the singularity <strong> 〉

2 ° singularity must be the center, but the center may not be the singularity, so the unintentional curve has no singularity.

3 ° at the singularity, the curve has a tangent along the gradual direction; while at the normal point, the curve has

X: Y =-(,) :(,) tangent, so that the tangent is unique at the normal point.

2,Nature:

The necessary condition for a 1 ° quadratic curve to have a singularity is = 0.

In fact, if the quadratic curve has a singularity (,),

Except equations have non-zero solutions (, 1)

Bytes = 0

Thoughts: = 0 is a sufficient condition for the quadratic curve to have a singularity? Why?

The necessary and sufficient condition for a 2 ° quadratic curve having a singularity is that it is a center quadratic curve, and its center is all over the quadratic curve,

In fact, "<strong" obviously

"Cosine>" sets the quadratic curve F (x, y) = 0 to have a singularity.

If the curve is a center quadratic curve, the unique center is also a singularity.

The region is centered on the curve;

If the curve is a line-center curve, it has a singularity equations.

Solution

Same Solution

Same Solution

Solution

And: =: Signature: =:

:=:=:

X + Y + = 0

The center of the vertex is a singularity, so that all the centers are on the curve.

 

Example:Returns the tangent l of the quadratic curve y²-4x-4y = 0.

(I) l point (3,-2 );

(Ii) l point (-1, 0 ).

Solution(I) The easy verification point (3,-2) is on the curve, and there is no singularity on the curve. The ∴ tangent equation is

-2y-2 (y-2)-2 (x + 3) = 0

That is, x + 2y + 1 = 0

(Ii) easy to verify (-) not on the curve,

Method 1: Set the tangent L and curve of (-) (,)

Then l: y-2 (Y +)-2 (x +) = 0

While (-1, 0) ε l records-2-2 (-1) = 0

That is, +-1 = 0

You²-4-4 = 0

Tangent =-= 2,-2 tangent L: 2y-2 (Y + 2)-2 (x-1) = 0

Or-2y-2 (y-2)-2 (x + 3) = 0 that

X + 1 = 0 or X + 2y + 1 = 0

Method 2:

Set (-1, 0) ELE. Me tangent L: then

△= [(-) X + (-) y] ²-phi (x, y) f (-) = 0

That is, [-2X-2Y] ²-4y² = 0

That is, x² + 2XY = 0 then X = 0 or X =-2Y

Between X: Y = or X: Y =-2:1 between l: Or l:

Method 3: Set the tangent l: X: Y = (x + 1): y

△= [(-) X + (-) Y] ²-phi (X, Y) F (-) = 0

Limit [(-) (x + 1) + (-) y] ²-4y² = 0

That is, [-1 (x + 1)-2y] ²-4y ² = 0

That is, (x + y + 1) ²-y² = 0 ∴ x + 1 = 0 or x + 2y + 1 = 0

3.The normal of the quadratic curve:

Definition:Set the tangent of the quadratic curve F (x, y) = 0, and the line perpendicular to the tangent is curved.

WiredNormal.

Method: If (,) is a normal point on the quadratic curve F (x, y ),

Normal l: