Analytic geometry. Textbook

at a point M0(x0; y0) of this circle may be written in the form

x0x + y0y a(x + x0) b(y + y0) + c = 0:

Proof. The point C(a; b) is the center of the circle. Using the same

! arguments as in the proof of the previous theorem we obtain that CM0 is

the normal vector of the tangent.

Fig. 2. Tangent to the circle, general case

!

Taking into account the equality CM0 = fx0 a; y0 bg we get the equation of the tangent:

(x0 a)(x x0) + (y0 b)(y y0) = 0:

We transform the equation of the tangent as follows:

(x0 a)x (x0 a)x0 + (y0 b)y (y0 b)y0 = 0; x0x + y0y ax by (x20 + y02 ax0 bx0) = 0:

Since the point M0 is on the circle we get that

x20 + y02 2ax0 2by0 + c = 0;

whence

x20 + y02 ax0 by0 = ax0 + by0 c;

and the equation of the tangent may be written in the form

x0x + y0y ax by (ax0 + by0 c) = 0;

or in the given above easy-to-remember form

x0x + y0y a(x + x0) b(y + y0) + c = 0:

Remark. To memorize equation of the tangent we note that the second degree terms x2 and y2 are replaced by x0x and y0y respectively and therst degree terms 2x and 2y are replaced by x + x0 and y + y0.

Theorem 5. The straight line Ax + By + C = 0 is the tangent to the circle (x x0)2 + (y y0)2 = r2 if and only if

p

jAx0 + By0 + Cj = r A2 + B2 :

Proof. A straight line is the tangent to the circle if and only if the distance of the center K of the circle to this line equals the radius of the circle.

K

Fig. 3. The tangency condition

In the case under consideration it means that

jAx0 + By0 + Cj

p = r: A2 + B2

Corollary. The straight line Ax + By + C = 0 is the tangent to the circle x2 + y2 = r2 if and only if r2(A2 + B2) = C2.

Proof. In this case x0 = 0, y0 = 0. Squaring the both sides of the condition given in the theorem we obtain the desired condition.

2.2The ellipse

2.2.1De nition and equation of the ellipse

Definition. The ellipse is the locus of the points of the plane, the sum of distances of which from two xed points of the plane, called the focuses of the ellipse, is a constant being larger than the distance between the focuses.

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

If the focuses coincide then r1 = r2 for each point M, the previous equation turns to the equation r1 = a and the ellipse becomes the circle.

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

Fig. 4. De nition of the ellipse: jF1Mj + jF2Mj = 2a

We rewrite the equation (4) in the coordinate form:

pp

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

Now there are implemented the following transformations. Transfer one of the radicals to the right side:

pp

(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 the radical to the left side and all the rest terms to the right side:

p

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

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

p

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

square the both sides:

a2((x c)2 + y2) = a4 2a2cx + c2x2;

remove the brackets:

a2x2 2a2cx + a2x2 + a2y2 = a4 2a2cx + c2x2;

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

rectly since a > c. Using this new constant, we rewrite the equation (5) in the following form: b2x2 + a2y2 = a2b2 or dividing the both sides of the last

We have proved that every point of the ellipse satis es the equation (6). Our transformations included squaring. Therefore, the resulting equation may become nonequivalent to the original condition (4)1).

Further we show that the initial condition (4) and the obtained equation (6) are equivalent, i.e. they describe the same curve.

Let M0(x0; y0) be an arbitrary point of the curve de ned be the equation (6), then

1) Squaring may result in the appearance of extra roots. For example in the simplest case when we square the equation x = 1, the new one x2 = 1 has the additional root x = 1.

Here we took into account that c2 = a2 b2. From the inequalities jx0j 6 a

and c < a we get that ac x0 < a, therefore a + ac x0 > 0 and from the equality jF1M0j2 = a + ac x0 2 we deduce that jF1M0j = a + ac x0.

The focal distance jF2M0j is found using the similar steps replacing c by c and we get that jF2M0j = a ac x0. Using the relations obtained for the focal distances jF1M0j and jF2M0j we get that

We have proved that any point satisfying the equation (6) has the property (4), so these conditions are equivalent. Equation (6) is called the canonical (or standard) equation of the ellipse.

In the following we assume that the ellipse under consideration is de ned by the equation (6).

The eccentricity of the ellipse is de ned by the equality

r

From the inequalities a > b, 0 6 c < a we get that 0 6 " < 1. In the case

" = 0 we have a = b and the ellipse turns into the circle.

The formulas for the focal distances may be rewritten in the following form: r1 = a + "x, r2 = a "x.

The coordinate axes are the axes of symmetry of the ellipse: if a point M(x0; y0) belongs to the ellipse, then the points M1( x0; y0) and

M3(x0; y0) also belong to this ellipse (symmetry relative the y-axis and the x-axis, respectively). The origin is the center of symmetry of the ellipse since with the point M(x0; y0) the ellipse contains the point M2( x0; y0) (see Fig. 5 below).

It can be proved that the ellipse has no more centers of symmetry and if the ellipse is not a circle (it means that " > 0) then it has no more axes of symmetry. If the ellipse is not a circle, then the intersection points of the ellipse with its axes of symmetry are called the vertices of the ellipse. These are the points ( a; 0) and (0; b).

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

In the rst quadrant we rewrite the equation of the ellipse in the form

q

y = b 1 xa22 , 0 6 x 6 a. We see that the function de ning the ellipse decreases monotonically and y(0) = b, y(a) = 0. The graph of ellipse is given in the Fig. 5.

Fig. 5. Ellipse and its symmetry,

the vertices of the ellipse are marked with circles

The values 2a and 2b are called respectively the major and minor axes of the ellipse. The values a and b are called the major and minor semiaxes respectively. Sometimes there is considered the equation (6) without assumption that a > b. If a < b then the values 2a and 2b become the minor and major axes respectively. The names of the semiaxes are changed similarly. In this case c2 = b2 a2, the focuses are at the points F1(0; c) and F2(0; c) and the eccentricity if found by the formula " = cb. In our further analysis we assume everywhere that a > b.

Let us discuss the geometrical properties of the ellipse de ned by the eccentricity. From the relations

p

we get that ab = 1 "2 . Therefore if " = 0 then a = b, the ellipse turns into the circle. If " ! 1 remaining less than 1 (this is denoted as " ! 1 0) then ab ! 0. This means that the ellipse becomes more and more at (or more and more oblong), see. Fig. 6.

"=0:4 "=0:5

"=0:6

"=0:7

"=0:8

"=0:9

"=0:95

"=0:99

"=0:999

Fig. 6. Ellipses with the same major axis (dashed line) and di erent eccentricities

2.2.2Directrices

Let us consider an ellipse that is not a circle.

Definition. The straight lines x = a" and x = a" are called the directrices of the ellipse.

The directrices are parallel to the axis Oy. Since " < 1 we get that a" > a and the directrices do not intersect the ellipse. The focus and the directrix,

being on the same side of the y-axis, are called corresponding.

Fig. 7. Directrices of the ellipse

Theorem 6. The point belongs to the ellipse if and only if the ratio of its focal distance to the distance of this point from the corresponding directrix equals the eccentricity of the ellipse.

2) Assume that for some point M(x; y) there holds the relation jF1Mj = ".

d1

We're just going to prove that the point M is on the ellipse, so we cannot use the formula for jF1Mj as in the previous case. Using the formulas

we get the equality

p

(x + c)2 + y2

= ":x + a"

We have proved that M is the point on the ellipse.

The case of the right focus and directrix is analyzed similarly.

2.2.3Tangent to the ellipse

Definition. A secant of a curve is a straight line intersecting the curve at a minimum of two distinct points.

In the case of the circle, we used the de nition of the tangent from the school math course. Now we de ne the tangent of the curve in the general case.

Definition. The tangent to a curve at its point A is the limit of the secants passing through the points A and B 6= A, when the point B tends to A.

Remark. In the case of a circle, this de nition is equivalent to the property used in the de nition of a tangent in the school math course.