Analytic geometry. Textbook

Proof. We consider the secant of the ellipse passing the point M0 and

Multiplying both sides of the equation (7) by the corresponding sides of the equality (8) we obtain:

Now we assume that M1 ! M0. Then x1 ! x0, y1 ! y0 and passing to the limit in (9) we get:

Canceling the equality by 2, removing the brackets and taking into account

Theorem 8. The straight line Ax + By + C = 0 is the tangent to the

ellipse x2 + y2 = 1 if and only if A2a2 + B2b2 = C2.

a2 b2

Proof. 1) Assume that the line Ax + By + C = 0 is the tangent to the given ellipse at its point M0(x0; y0). Then the same line is de ned by the equation xa02x + yb02y = 1. Therefore

From here we get that C 6= 0 and x0 = CA a2, y0 = BC b2. Since M0 is the point of the ellipse we get:

2) Assume that there holds the equality A2a2 +B2b2 = C2. Then C 6= 0, since otherwise A2a2 + B2b2 = 0 and A = 0, B = 0 which is impossible

for the equation of the straight line. Let M0(x0; y0) be the point with the coordinates x0 = CA a2, y0 = BC b2 (we write out these equalities from the previous part of the proof). Substituting these coordinates into equation of the ellipse and using the relation A2a2 + B2b2 = C2 we obtain that M0 is the point of the ellipse:

Using the previous theorem we get that the tangent to the ellipse at the point M0 may be de ned by the equation

and after simpli cation we get the equation Ax + By + C = 0.

We obtain that the line de ned by the equation Ax + By + C = 0 is the

tangent.

Remark. It is easy to prove the following additions to the previous

theorem.

1) the line is the secant of the ellipse if and only if A2a2 + B2b2 > C2;

2) the line does not intersect the ellipse if and only if A2a2 + B2b2 < C2.

2.2.4Parametric equations of the ellipse

Theorem 9. Ellipse with the equation x2 + y2 = 1 may be de ned by the

a2 b2

parametric equations

Then there exist such a value t, 0 6 t < 2 that xa0 = cos t, yb0 = sin t. We get that x0 = a cos t, y0 = b sin t for some t 2 [0; 2 ).

2.3The hyperbola

2.3.1De nition and equation of the hyperbola

Definition. The hyperbola is the locus of the points on the plane, for which absolute value of di erences of their distances from two xed points of the plane, called the focuses of the hyperbola, is a positive constant being less than the distance between the focuses.

As in the case of ellipses, we denote the focuses of the hyperbola by F1,

F2. Let jF1F2j = 2c. The distances r1 and r2 from any point M(x; y) of the hyperbola to the points F1 and F2 are called the focal distances. According the de nition of the hyperbola, the absolute value of the focal distances is a constant value. Let us denote it by 2a. It is assumed that 0 < 2c < 2a, i.e. 0 < c < a. So the hyperbola is de ned by the equation

y

M

Let us consider the hyperbola with the parameters a and c de ned above

and the focuses F1, F2. We introduce the coordinate system de ned by the following conditions: the midpoint of the segment F1F2 is the origin, the x-axis passes through the points F1 and F2 and is directed from F1 to F2. In this coordinate system the focuses have the following coordinates: F1( c; 0),

F2(c; 0). Then the focal distances take the values

pp

r1 = (x + c)2 + y2 ; r2 = (x c)2 + y2 ;

and we rewrite the equation (10) in the coordinate form:

pp

j (x + c)2 + y2 (x c)2 + y2 j = 2a

or

pp

(x + c)2 + y2 (x c)2 + y2 = 2a

We implement the following transformations. Transfer the second radical to the right side:

p p

(x + c)2 + y2 = 2a + (x c)2 + y2 ;

square the both sides:

p

(x + c)2 + y2 = 4a2 4a (x c)2 + y2 + (x c)2 + y2;

transfer all the terms except the radical from the right side to the left one:

p

(x + c)2 + y2 4a2 y2(x c)2 + y2 = 4a (x c)2 + y2 ;

remove the brackets, collect similar terms and cancel by 4:

p

xc a2 = a (x c)2 + y2 ;

square both sides:

x2c2 2acx + a4 = a2((x c)2 + y2);

remove the brackets:

x2c2 2acx + a4 = a2x2 2a2cx + a2c2 + a2y2;

56 Chapter 2. Second order curves on the plane

gather the terms containing x or y in the left side and the constant values

p

We introduce the new constant b = c2 a2 which is de ned correctly since

c > a. Using this constant we rewrite the equation (11) in the following form: b2x2 a2y2 = a2b2, dividing the both sides of the last equation by a2b2

We obtain that every point of the hyperbola satis es the equation (12). Since our transformations included squaring, the resulting equation may be nonequivalent to the original condition (10).

Therefore x20 > a2, jx0j > a.

In order to prove that the point satisfying the equation (12) also satis es the initial condition (10) we consider the following two cases. Here we take

valid.

We have proved that any point satisfying the equation (12) has the property (10), so these conditions are equivalent. Equation (12) is called

the canonical (or standard) equation of hyperbola.

In the following, we suppose that the hyperbola under consideration is

de ned by the equation (12).

The eccentricity of the hyperbola is de ned by the equality

Since c > a we get that " > 1. The formulas for the focal distances may be rewritten in the following form: r1 = j"x + aj, r2 = j"x aj.

The coordinate axes are the axes of symmetry of the hyperbola: if a point M(x0; y0) belongs to the hyperbola, then the points M1( x0; y0) and M3(x0; y0) also belong to this hyperbola (symmetry relative y-axis and x-axis, respectively). The origin is the center of symmetry of the hyperbola since with the point M(x0; y0) the hyperbola contains the point

M2( x0; y0) (see Fig. 9). It can be proved that the hyperbola has no more centers of symmetry and no more axes of symmetry. The intersection points of the hyperbola with the x-axis are called the vertices of the hyperbola. These are the points ( a; 0).

In order to plot the hyperbola, we may use its symmetry. Due to the symmetry with respect to the coordinate axes, it is su cient to plot the hyperbola in the rst quadrant and then extend it to the rest of the coor-

dinate plane. In the rst quadrant, the equation of the hyperbola may be

q

rewritten if the form y = b xa22 1 , x > a. This function is monotonically increasing on the set [a; +1).

Fig. 9. Hyperbola and its symmetry,

the vertices of the hyperbola are marked with circles

We consider the line ` with the equation y = ab x and nd the distance d of the point M(x; y) of the hyperbola from this line. Rewriting equation of the line in the form bx ay = 0 we get:

y

x

Fig. 10. Hyperbola and its asymptotes

From the obtained formula we deduce that d ! 0 as x ! +1. From the p

obvious inequality ab x2 a2 < ab x we get that the points of the hyperbola in the rst quadrant lie below the line y = ab x. This line is called the asymptote of the hyperbola. The relative position of the hyperbola and its asymptote in the other quadrants is easily deduced from the case of therst quadrant analyzed above and symmetry considerations. The second asymptote of the hyperbola is de ned by the equation y = ab x. Equations of the asymptotes may be written in the form xa yb = 0, xa + yb = 0. It may be proved that the hyperbola has no other asymptotes, that is, straight lines that approach the hyperbola at in nity.

The parts of the hyperbola being on opposite sides of the y-axis are called its branches. The formulas for the focal distances may be rewritten in the following form:

r1 = jMF1j = "x + a; r2 = jMF2j = "x a

for a point M on the right branch of the hyperbola,

r1 = jMF1j = "x a; r2 = jMF2j = "x + a

for a point M on the left branch of the hyperbola.