Lecture 2. Hyperbola and parabola
In this lecture, we will continue to study second-order curves and consider in detail two lines called hyperbola and parabola
Let's start with the concept of hyperbola. From your school maths course, you know that a hyperbola is a graph of a function called inverse proportionality, which is given by the following equation: y=k/x. It turns out that the graph of such a function gives us a frequent case of a hyperbola, which is called equiangular. We will now give a general concept of a hyperbola and then compose its equation in some specially selected coordinate system.
A hyperbola is a set of points in the plane, the module of the distance difference from each to 2 fixed points is a constant value. Two fixed points are called foci of the hyperbola. We will denote them in the same way as for the ellipse, with the letters F1 and F2. The distance between the foci is called the inter-focus distance. The length of this segment F1F2 is denoted by 2C: | F1F2 |=2c. Choose an arbitrary point M. In order for it to belong to a hyperbola, the following equation must be true: ||MF1|-|MF2||=2a by virtue of this definition (that is, the module of the distance difference must be equal to a constant number, let's denote this constant number by 2a). The distances from the point M to the foci themselves are called the focal radii of the point M, which lies on the hyperbola.
Let's point out some points in the figure, that is, imagine what this figure looks like. It consists of two parts, which are called hyperbola branches. If we change the location of the foci, namely, let's try to bring them closer together, then, in this case, the two branches of the hyperbola will approach each other and in the extreme case will merge into two intersecting lines. The intersection point is two coinciding foci.
Now consider the coordinate system in which we derive the hyperbola equation. Just as for an ellipse, we take the point O as the origin – the middle of the segment F1F2,, the axis of the abscissa is directed along the foci F1 and F2, the axis of the ordinates is perpendicular to the axis of the abscissa and comes through the origin, since we have a Cartesian coordinate system. So, the resulting coordinate system is called the canonical system. The axes of this system are the axes of symmetry of the hyperbola, that is, the hyperbola is symmetric with respect to these axes. The foci F1 and F2receive certain coordinates in this system: they lie on the abscissa axis at points -c and c. the Hyperbola intersects the abscissa axis at points a and -a.
These points are called the real vertices of the hyperbola, and the parameter a itself is called the real semi-axis. Just as for the ellipse, we define the parameter b, here it is determined by the equality that you see on the screen. Let's put points b and -b on the ordinate axis. The resulting points are called imaginary vertices of the hyperbola, it does not pass through them. The parameter b itself is called the imaginary half-axis. If you draw straight lines parallel to the coordinate axes through these four points, you will get a rectangle called the reference rectangle for this hyperbola. Next, we consider the lines that contain the diagonals of this rectangle, these lines are called hyperbola asymptotes.
Note that the hyperbola approaches such straight lines indefinitely, but does not intersect them. The asymptotes can be given by these equations (see the video), since the lengths of the sides of the reference rectangle are 2a and 2b. If we take an arbitrary point M on the hyperbola and consider the corresponding focal radii, they can be calculated using the formulas shown on the slide for any point M lying on the hyperbola. So, the basic concepts are defined.
Now consider the hyperbola equation in the selected coordinate system. It is derived in the same way as we obtained the ellipse equation. We choose a point M on the hyperbola, given the coordinates of the foci, the coordinates of the point M that satisfy the equality by which the hyperbola is defined. We get this equation with the variables x and y (see the video). Next, we perform the transformation: we square it, simplify it, and as a result we get the following equality. Remembering how we defined the parameter b by performing a number of other transformations, we reduce this equality to the following (see the video). The resulting equation resembles that of an ellipse. The only difference is that for the ellipse we had the sum offractions in the left part, here is the difference. Pay attention to this. The resulting equation is called the canonical equation of the hyperbola (see video).
Just as for an ellipse, you can introduce the concept of a tangent. That is, we fix a point on the hyperbola, consider all possible secants. The limiting position of the secant is called the tangent to the hyperbola at a given point. For each point of the hyperbola, the tangent can be set by the following equation (see video). For example, consider one of the vertices of the hyperbola through which it passes, namely the vertex with coordinates (a; 0). Substituting these coordinates into the equation, we get that at this vertex the hyperbola touches the line given by the equality x=a. Thus, the hyperbola touches the vertical sides of the reference rectangle.
An optical property is true for the hyperbola that is the following: if we consider an arbitrary focus and place a light source there, all rays emerging from the source are reflected from the nearest branch of the hyperbola will get the direction, which coincides with the direction of the vector going from the second focus point of reflection.
Just as for an ellipse, we can introduce the concept of eccentricity – this is the ratio of the parameter c to a. The definition is exactly the same, but if for an ellipse the eccentricity was enclosed in the interval (0; 1), then for a hyperbola this characteristic takes a value greater than 1, since the parameter a is less than c.
Let's consider some examples of hyperballs, and for each example we will indicate the corresponding value of the eccentricity. From the figures, we see that the greater the value of the eccentricity, the more the branches of the hyperbola are stretched along the ordinate axis. The fact is that with an increase in the ratio of c to a, the ratio of b to a increases, that is, the parameter b begins to increase indefinitely with respect to a.
Now let's think about the graph of inverse proportionality, with which we started this lecture. Let's rotate the points of the plane by an angle of 45⁰ clockwise around the point O. In this case, each point M will pass to some new point M`. Consider the inverse proportionality graph given by y=k/x, where k>0, and perform the specified rotation transformation. In this case, each point M of the hyperbola will move to some new point M`. After performing this transformation with each point, the two branches of the inverse proportionality graph will move to the line indicated on the slide.
Using the rotation formulas, you can show that each point of the specified line satisfies the equation written on the slide, which, as we see, is the equation of a hyperbola, while this hyperbola has the same semi-axes. Hyperbolas with equal semi-axes are called equiangular. Thus, we can say that the inverse proportionality graph is an equiangular hyperbola.
Now we take an arbitrary equiangular hyperbola given by the above equation and perform the compression-to-abscissa transformation already considered earlier with the given coefficient b/a. In this case, each point of the taken equiangular hyperbola will move to some new point M`, and the coordinates of the new point and the original ones are connected by the specified equalities (see the video). After performing this compression transformation, we will get some new line. Substituting these equalities connecting the coordinates of the points into the equation of an equiangular hyperbola with the semiaxis a, we get the equation of an arbitrary hyperbola with the semiaxis a and b. Thus, an arbitrary hyperbola can be obtained from an equiangular hyperbola using a compression transformation.
Now let's move on to another second-order curve called a parabola. Again, you know from a school maths course that a parabola is a graph of a quadratic function given by the equation you see on the screen, and it is known that the graph of the derivative of a quadratic function can be obtained from the graph of the function y=kx2 by parallel transfer along the coordinate axes. Now we will give a general definition of a parabola, then derive its equation and make sure that indeed a parabola is a graph of a quadratic function.
So, a parabola is a set of points in the plane equidistant from a fixed point F and some straight line l, which does not contain this point. The fixed point F is called the parabola focus, and the fixed line l is called the directrix. Let's make a drawing, fix the point and the straight line. According to the definition, we have the following characteristic equality (see the video), any point M of the parabola must have this property, that is, the distance MF must be equal to the distance from this point to the line l. We introduce one parameter p, called the fatal parameter of the parabola, it is equal to the distance from the point F to the directrix l.
Let's draw several points lying on the parabola satisfying the specified relation. So, we get the following line (see video). If we now change the value of the parameter p, the parabola will change. In particular, if the value of p tends to zero, then the distance between the focus and the line will also tend to 0, the focus will try to take a position on this line, and the parabola will degenerate into a ray.
Now determine the equation of a parabola. Let's define the parabola by the equation. To do this, we introduce the canonical coordinate system as follows: the origin is the middle of the perpendicular dropped from the focus to the directrix, the abscissa axis is a straight line OF in the direction of the focus, and the ordinate axis is again perpendicular to the abscissa axis and comes through the point O. The point O belongs to a parabola, it is called a vertex, and the abscissa axis is the axis of symmetry of the parabola, since the parabola is symmetric with respect to this axis. The resulting coordinate system is called the canonical system. In this system, the focus has coordinates (p/2; 0), and the directrix is given by the following equation: x=-p/2, where p is the focal parameter.
Now we derive the equation. Take an arbitrary point lying on a parabola. Let's remember how this point is defined: it satisfies the specified equality (see the video). Let's express the distance specified in the equation in terms of the coordinates of the point M and the coordinates of the focus. We get the following equation (see the video). Let's square it and with the help of simple transformations, this equation will take this form. The resulting equality is called the canonical parabola equation (see video).
Now let's express the variable x from this equality, we get that x is a function of y, and we have exactly the quadratic dependence of x on y. If you re-assign the variables x and y, that is, change the coordinate axes mutually, then the dependence will take the form: y=kx2. That is, we get the quadratic dependence of y on x. Thus, the graph of a quadratic function is a parabola.
Consider an arbitrary parabola in the canonical coordinate system. Take an arbitrary point, consider the tangent at that point, which is defined in the same way as for an ellipse and for a hyperbola, and write down the equation of the tangent at that point. It has the form indicated on the slide: (x0; у0) is the coordinate of the point, p is the focal parameter (see the video).
Consider the optical property of a parabola: if a radiation source is placed in its focus, then the rays directed from this point, reflected from the parabola, will take a direction parallel to the axis of the abscissa, and the direction coinciding with the axis of the abscissa. At the same time, the opposite is also true: if you direct a stream of rays parallel to the axis of symmetry, then these rays, reflected from the parabola, will converge in its focus.
A parabola is a flat shape, if we rotate it around its axis, we get a spatial shape called a paraboloid of rotation. This spatial figure has exactly the same optical property.
In conclusion, we will summarize some results. We will try to remember all the considered lines of the second order, establish some relationship between them.
First, we note that all three specified lines - ellipse, hyperbola and parabola, can be obtained from some surface by the method of sections, namely, let's consider the following surface (see video). Take an arbitrary circle and a point that does not lie in the plane of the circle. For each point of the circle, we draw a line passing through the initially selected point, and the set of all such lines form a certain surface, which is called a cone or a conical surface. Consider three figures, each of which produces a certain second-order line - an ellipse, a parabola, and a hyperbola. In particular, these lines can degenerate into a point, a straight line, or two straight lines.
When we defined a parabola, we introduced the concept of a directrix. It turns out that this concept can be defined for an ellipse and for a hyperbola by the specified equality. For a parabola, the directrix is such a line that any point of the parabola is equidistant from that line and from the focus. For an ellipse, we can consider 2 straight lines given by the above equations (see video). Similarly for a hyperbola. So, the concept of directrix can be introduced for all three lines. The following theorem is true: for any line l and a point F that does not lie on the line, and the number ε > 0, the specified equality works (see the video).
In this case, for ε < 1 we get an ellipse other than a circle, if ε = 1 we get a parabola, since equality is exactly defined by a parabola, if ε > 1 we get a hyperbola. For an ellipse and a hyperbola, ε, as we know, is called an eccentricity. Thus, for a parabola, we can assume that the eccentricity is 1, for an ellipse less than 1, for a hyperbola more than 1.
If we introduce the concept of a focal parameter for an ellipse and a hyperbola (we introduced this concept only for a parabola), that is, we generalize this concept for two other lines of the second order, then we can describe all three non-degenerate curves of the second order by a single equation, which is indicated on the slide.
Let us consider a few figures in order to visualize this theorem. Let ε = 0, in this case we have a circle, then we begin to increase the value of ε in the interval (0; 1). We get a different position of the ellipse. Further, if ε turns into 1, we have an ellipse transformed into a parabola, further increasing ε, that is, considering it greater than 1, we will already get various hyperbolas. As we know, by increasing the eccentricity, the branches of the hyperbola begin to stretch along the ordinate axis.