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
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.
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
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
is a condition that the line y = kx + b touches the circle x² + y² = r²
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
The equation of the normal of the circle x² + y² = r² at the point M (x₁, y₁) is
and the equation of the normal of the circle
(x-p) ² + (y-q) ² = r²
at the point M (x₁, y₁) is
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
is called the central or canonical equation of an ellipse.
Fig.8
The segment A₁A₂ is the major axis of the ellipse and
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
The following relation applies to the axles and eccentricity
2.2. Ellipse directories
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:
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
is called the central or canonical equation of hyperbole.
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 F₁ and 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:
where it comes from
3.2. Hyperbole Directories
Hyperbola has two directorates and their equations are:
3.3. Asymptotes of hyperbole
You do, determined by the equations
are asymptotes of hyperbole
Fig.9
3.4. Mutual position of straight and hyperbole
Rights and hyperbole have the same reciprocal relations as rights and circles.
The relation
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:
and the equation of the normal:
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).
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.
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
and the equation of the normal:
