Second order curves

The equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 is a general quadratic equation with two unknowns.

The curve determined by a quadratic equation with two variables is called a quadratic curve.

Second-order curves, for example, are: circle, ellipse, hyperbole, and parabola.

1. CIRCLE

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The equation:

is an equation of a circle with center at point C (p, q) and radius r (Fig.7).

The equation x² + y² = r² is called the central equation of a circle.

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Fig. 7

1.1. Mutual relation of rights and circle

Let be the equation of the circle

(x-p) ² + (y – q) ² = r²

and the equation of rights

y = kx + b.

If the system equations

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there are two solutions, then the line intersects the circle; if there is one solution-it touches the circle and if there is no solution-the line and the circle have no common points.

The equation

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is a condition that the line y = kx + b touches the circle x² + y² = r²

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The equation

is a condition that the line y = kx + b touches the circle

(x – p) ² + (y – q) ² = r²

1.2. Equation of tangent and perpendicular to circle

The equation of the tangent of the circle x² + y² = r² at the point M (x₁, y₁) is

x₁x + y₁y = r²

If the equation of the circle is given in the form

(x-p) ² + (y – q) ² = r²

then the equation of the current at the point M (x₁, y₁) is

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The equation of the normal of the circle x² + y² = r² at the point M (x₁, y₁) is

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and the equation of the normal of the circle

(x-p) ² + (y-q) ² = r²

at the point M (x₁, y₁) is

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2. Ellipse

An ellipse is a set of points in a plane where the sum of the distances of any of its points to two given points is constant (fig. 8).

The equation

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is called the central or canonical equation of an ellipse.

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Fig.8

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The segment A₁A₂ is the major axis of the ellipse and

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The segment B₁B₂ is the small axis of the ellipse and

The points F₁ and F₂ are foci of the ellipse and

F̄₁̄F̄₂ = 2c

The half-axes a and b and the half-focal distance s meet the following relations:

a> c and a²-c² = b²

2.1. Ellipse eccentricity

The eccentricity of the ellipse is

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The following relation applies to the axles and eccentricity

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2.2. Ellipse directories

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The ellipse directories are two normal lines of lines on which lies its major axis, symmetrically with respect to the center of the ellipse and at a distance a / ε from it. The equations of the directories are:

2.3. Mutual relationship of rights and ellipse

Rights and ellipses have the same reciprocal relations as
straight and circular.

Let be the equation of the ellipse b²x² + a²y² = a²b² and the equation of the line у = kx + n. The relation

a²k² + b² – n² = 0 or a²k² + b² = n²

is a condition that the line u = kx + n touches the ellipse
b²x² + a²y² = a²b²

2.4. Tangent equation and ellipse normal

The equation of the tangent of the ellipse b²x² + a²y² = a²b² at the point M (x₁y₁) reads:

b²x₁x + a²y₁y = a²b²

The equation of the ellipse normal b²x² + a²y² = a²b² at the point M (x₁, y₁) reads:

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3. HYPERBOLA

Hyperbola is a set of points on a plane such that the difference in the distances of WHICH was its point up to two given points is constant (fig. 9).

The equation

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is called the central or canonical equation of hyperbole.

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Fig.9

The segment A₁A₂ is the real axis of the hyperbola and A₁A₂ = 2a (a – half axis). The segment B₁B₂ is the imaginary axis of the ellipse and B₁B₂ = 2b (b-half axis). The points Fand F₂ are foci of the ellipse and F₁F₂ = 2c (s – half-focal length).

The following relations apply to the half-axes a and b and the half-focus distance:

a <c and c²- a² = b²

3.1. Eccentricity of hyperbole

The quotient s: a from the half-focal length and the real half-axis of the hyperbola is called the eccentricity of the hyperbola and is denoted by ε.

Since s> a, e> 1 follows. In the case of hyperbole, the following applies:

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where it comes from

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3.2. Hyperbole Directories

Hyperbola has two directorates and their equations are:

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3.3. Asymptotes of hyperbole

You do, determined by the equations

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are asymptotes of hyperbole

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337-f3 Fig.9

3.4. Mutual position of straight and hyperbole

Rights and hyperbole have the same reciprocal relations as rights and circles.
The relation

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that is

a²k² – b² = n²

is a condition that the line u = kx + n touches the hyperbola b²x² – a²y² = a²b²

3.5. Tangent equation and hyperbola normal

The equation of the tangent of hyperbola b²x² – a²y² = a²b²

at any of its points M (x₁, y₁) is:

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and the equation of the normal:

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4. PARABLE

A parabola is a set of points in a plane that are equidistant from one constant point and one constant line.

The normal distance from the focus to the direction is usually denoted by p and is called the parabola parameter (fig. 10).

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Fig. 10

The equation y² = 2 p x is called the canonical equation of a parabola.

The point O (0.0) is the vertex of the parabola. The focus has coordinates p / 2 and O, ie.

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4.1. Mutual relation of rights and parabola

The relation р = 2kn is a condition for the line y = kx + n to touch the parabola y² = 2рх.

2.4.2. Tangent equation and normal to parabola

The tantrum equation of the parabola y² = 2rx at its point M (x₁, y₁) is

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and the equation of the normal:

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