Definition 7.2. The locus of points in a plane for which the difference between the distances to two fixed points is a constant is called hyperbole.

Remark 7.2. Speaking of the difference in distances, they mean that a smaller distance is subtracted from a larger one. This means that in fact, for a hyperbola, the modulus of the difference in distances from any of its points to two fixed points is constant. #

The definition of a hyperbola is similar to the definition ellipse. The difference between them is only that for a hyperbola the difference of distances to fixed points is constant, and for an ellipse - the sum of the same distances. Therefore, it is natural that these curves have much in common both in properties and in the terminology used.

Fixed points in the definition of a hyperbola (we denote them by F 1 and F 2) are called foci of hyperbole. The distance between them (we denote it by 2s) is called focal length, and the segments F 1 M and F 2 M, connecting an arbitrary point M on the hyperbola with its foci, - focal radii.

The form of the hyperbola is completely determined by the focal length |F 1 F 2 | = 2с and the value of the constant value 2а, equal to the difference of the focal radii, and its position on the plane - the position of the foci F 1 and F 2 .

It follows from the definition of a hyperbola that, like an ellipse, it is symmetrical with respect to the straight line passing through the foci, as well as with respect to the straight line that divides the segment F 1 F 2 in half and is perpendicular to it (Fig. 7.7). The first of these axes of symmetry is called the real axis of the hyperbola, and the second - her imaginary axis. The constant a involved in the definition of a hyperbola is called the real semiaxis of the hyperbola.

The middle of the segment F 1 F 2 connecting the foci of the hyperbola lies at the intersection of its axes of symmetry and therefore is the center of symmetry of the hyperbola, which is simply called the center of the hyperbola.

For a hyperbola, the real axis 2a must be no greater than the focal distance 2c, since for the triangle F 1 MF 2 (see Fig. 7.7) the inequality ||F 1 M| - |F 2 M| | ≤ |F 1 F 2 |. The equality a = c holds only for those points M that lie on the real axis of symmetry of the hyperbola outside the interval F 1 F 2 . Discarding this degenerate case, we further assume that a

Hyperbola equation. Let us consider some hyperbola on the plane with foci at the points F 1 and F 2 and the real axis 2a. Let 2c be the focal length, 2c = |F 1 F 2 | > 2a. According to Remark 7.2, the hyperbola consists of those points M(x; y) for which | |F 1 M| - - |F 2 M| | = 2a. Let's choose rectangular coordinate system Oxy so that the center of the hyperbola is at origin, and the foci were located on abscissa(Fig. 7.8). Such a coordinate system for the considered hyperbola is called canonical, and the corresponding variables - canonical.

In the canonical coordinate system, the foci of the hyperbola have coordinates F 1 (c; 0) and F 2 (-c; 0). Using the distance formula between two points, we write the condition ||F 1 M| - |F 2 M|| = 2a in coordinates |√((x - c) 2 + y 2) - √((x + c) 2 + y 2)| \u003d 2a, where (x; y) are the coordinates of the point M. To simplify this equation, we get rid of the modulus sign: √ ((x - c) 2 + y 2) - √ ((x + c) 2 + y 2) \u003d ±2a, move the second radical to the right side and square it: (x - c) 2 + y 2 \u003d (x + c) 2 + y 2 ± 4a √ ((x + c) 2 + y 2) + 4a 2 . After simplification, we get -εx - a \u003d ± √ ((x + c) 2 + y 2), or

√((x + c) 2 + y 2) = |εx + a| (7.7)

where ε = c/a. We square a second time and again bring similar terms: (ε 2 - 1) x 2 - y 2 \u003d c 2 - a 2, or, given the equality ε \u003d c / a and setting b 2 \u003d c 2 - a 2,

x 2 / a 2 - y 2 / b 2 \u003d 1 (7.8)

The value b > 0 is called imaginary semiaxis of the hyperbola.

So, we have established that any point on a hyperbola with foci F 1 (c; 0) and F 2 (-c; 0) and a real semi-axis a satisfies equation (7.8). But we must also show that the coordinates of points outside the hyperbola do not satisfy this equation. To do this, we consider the family of all hyperbolas with given foci F 1 and F 2 . This family of hyperbolas has common axes of symmetry. It is clear from geometric considerations that each point of the plane (except for the points lying on the real axis of symmetry outside the interval F1F2 and the points lying on the imaginary axis of symmetry) belongs to some hyperbola of the family, and only one, since the difference in the distances from the point to the foci F 1 and F 2 changes from hyperbole to hyperbole. Let the coordinates of the point M(x; y) satisfy equation (7.8), and let the point itself belong to the hyperbola of the family with some value ã of the real semiaxis. Then, as we have shown, its coordinates satisfy the equation Therefore, a system of two equations with two unknowns

has at least one solution. By direct verification, we make sure that for ã ≠ a this is impossible. Indeed, eliminating, for example, x from the first equation:

after transformations, we obtain the equation

which, for ã ≠ a, has no solutions, since . So, (7.8) is an equation of a hyperbola with a real semi-axis a > 0 and an imaginary semi-axis b = √ (с 2 - a 2) > 0. It is called the canonical equation of the hyperbola.

Type of hyperbola. In its form, the hyperbola (7.8) differs markedly from the ellipse. Taking into account the presence of two axes of symmetry of the hyperbola, it is enough to construct that part of it that is in the first quarter of the canonical coordinate system. In the first quarter, i.e. for x ≥ 0, y ≥ 0, the canonical equation of the hyperbola is uniquely resolved with respect to y:

y \u003d b / a √ (x 2 - a 2). (7.9)

The study of this function y(x) gives the following results.

The domain of the function is (x: x ≥ a) and in this domain it is continuous as a complex function, and at the point x = a it is continuous on the right. The only zero of the function is the point x = a.

Let's find the derivative of the function y (x): y "(x) \u003d bx / a √ (x 2 - a 2). From this we conclude that for x> a the function is monotonically increasing. In addition, , which means that at the point x = a of the intersection of the graph of the function with the x-axis there is a vertical tangent. The function y(x) has a second derivative y" = -ab(x 2 - a 2) -3/2 for x> a, and this derivative is negative. Therefore, the graph of the function is convex upwards, and there are no inflection points.

This function has an oblique asymptote, which follows from the existence of two limits:

The oblique asymptote is described by the equation y = (b/a)x.

The study of the function (7.9) allows us to construct its graph (Fig. 7.9), which coincides with the part of the hyperbola (7.8) contained in the first quarter.

Since the hyperbola is symmetrical about its axes, the entire curve has the form shown in Fig. 7.10. A hyperbola consists of two symmetrical branches located at different

side of its imaginary axis of symmetry. These branches are not bounded on both sides, and the lines y = ±(b/a)x are simultaneously asymptotes of both the right and left branches of the hyperbola.

The axes of symmetry of the hyperbola differ in that the real one intersects the hyperbola, and the imaginary one, being the locus of points equidistant from the foci, does not intersect (which is why it is called imaginary). Two points of intersection of the real axis of symmetry with the hyperbola are called the vertices of the hyperbola (points A (a; 0) and B (-a; 0) in Fig. 7.10).

The construction of a hyperbola along its real (2a) and imaginary (2b) axes should begin with a rectangle centered at the origin and sides 2a and 2b parallel, respectively, to the real and imaginary axes of symmetry of the hyperbola (Fig. 7.11). The asymptotes of a hyperbola are continuations of the diagonals of this rectangle, and the vertices of the hyperbola are the points of intersection of the sides of the rectangle with the real axis of symmetry. Note that the rectangle and its position on the plane uniquely determine the shape and position of the hyperbola. The ratio b/a of the sides of the rectangle determines the degree of compression of the hyperbola, but instead of this parameter, the eccentricity of the hyperbola is usually used. The eccentricity of a hyperbola called the ratio of its focal length to the real axis. The eccentricity is denoted by ε. For the hyperbola described by equation (7.8), ε = c/a. Note that if ellipse eccentricity can take values ​​from a half-interval )