Analytic geometry. Textbook

2) Similar reasoning leads to the following statement: the straight line

Ax + By + C = 0 is parallel to the y-axis if and only if B = 0.

3) If C = 0 then the equation of the straight line takes the form Ax +

By = 0 and the line passes through the origin since the values x = 0, y = 0

satisfy this equation.

Conversely, if the line passes through the origin, then substituting the

values x = 0, y = 0 in the equation of the line, we obtain that C = 0.

4) Using the previous results, we get that the relations A = 0, C = 0 take place if and only if the line is parallel to the x-axis and passes through the origin. It means that the line coincides with the x-axis. From similar arguments we get that the condition B = 0 and C = 0 is valid if and only

if the line coincides with the y-axis.

We collect the information given above in the following table.

1.2.3Distance of a point from a line

Theorem 2. Let ` be a line on the coordinate plane de ned by the equation Ax + By + C = 0 and let M1(x1; y1) be some point on this plane. Then the the distance d of the point M1 from the line ` is found by the formula

jAx1 + By1 + Cj d = p :

A2 + B2

Proof. We consider separately the cases when the point M1 is on the straight line ` or is not.

Let us assume rst that M1 is not the point of the line `.

22 Chapter 1. Straight lines on the plane

collinear. Therefore the angle ' between these vectors takes one of the values 0, , thus cos ' = 1 (these two cases are shown in Fig. 16). From the relations

= Ax1 + By1 (Ax0 + By0):

The point M0 is on the line `, therefore

Ax0 + By0 + C = 0; Ax0 + By0 = C

!

and ~n M0M1 = Ax1 + By1 + C. Using this equality and the equality j~nj = p

A2 + B2 we obtain from (8) the nal result:

jAx1 + By1 + Cj d = p :

A2 + B2

If the line ` passes through the point M1, then the last equality remains true since in this case d = 0 and Ax1 + By1 + C = 0.

Corollary 1. A straight line divides the plane into two half-planes. If a straight line is de ned by the equation Ax + By + C = 0 then one of these half-planes is de ned by the inequality Ax + By + C > 0 and the other one is de ned by the inequality Ax + By + C < 0.

Remark. The corollary immediately follows from the proof of the theorem. The \positive" and \negative" half-planes are nor de ned by the straight line itself, they depend on the choice of the equation of the the straight line. The equations Ax + By + C = 0 and Ax By C = 0 de ne the same straight line but the corresponding half-planes change their places. It is easy to check that the normal vector of the straight line de ned by its equation is directed into the \positive" half-plane. We considered here open half-planes. In this case it means that they do not contain the dividing straight line. It is also possible to consider the closed half-planes which contain the given straight line. For these half-planes all the facts given above stay true if we replace the strict inequalities by non-strict.

Corollary 2. The distance between the parallel straight lines de ned by the equations Ax + By + C1 = 0 and Ax + By + C2 = 0 may be found by the formula

Proof. Let us take an arbitrary point M0(x0; y0) on the rst straight line. The distance d equals the distance between this point and the second straight line. Therefore

in (9) the sum Ax0 + By0 by C1 we get the desired formula.

1.2.4Intercepts

The abscissas of the points in which the graph of a curve cuts the x- axis are called the x-intercepts of the graph. The ordinates of the points in which the graph cuts the y-axis are called the y-intercepts of the graph. Taking into account that the ordinate of every point on the x-axis is 0 and the abscissa of every point on the y-axis is 0, we obtain the following rules which are valid not only in the case of straight lines considered here, but also in the case of more general lines on the plane.

To nd the x-intercepts of the graph of an equation, we put y = 0 and solve the equation with the unknown x.

To nd the y-intercepts of the graph of an equation, we put x = 0 and solve the equation with the unknown y.

If the straight line is parallel to the y-axis it has the single x-intercept, similarly for the case of x-axis. If the straight line is not parallel to each of axes, then the both intercepts are uniquely de ned.

Assume that the straight line is de ned by the equation Ax+By+C = 0 and B 6= 0. It means that line is not parallel to the y-axis and therefore intersects this axis at a single point. Expressing the coordinate y from the given equation we get: By = Ax C, y = BA x BC or y = kx + b, where we denote k = BA , b = BC . It is said that such an equation has the slope

(or slope-intercept) form. Putting the value x = 0 into the equation we get y = b. So the y-intercept is b.

Fig. 17. Straight lines with positive and negative slopes and corresponding angles of inclination

The coe cient k is called the slope of the straight line. This value determines the angle of inclination of the line to the positive direction of x- axis. This angle is counted from the x-axis in the counterclockwise direction. If k > 0 this angle is acute and the straight line goes up when moving from left to right. If k < 0 this angle is obtuse and the straight line goes down when moving from left to right. In the case k = 0, the equation takes the form y = b and the straight line is parallel to the x-axis (the straight line is \horizontal"). The angle of inclination of the line may be found by the formula = arctan k if k > 0 and = + arctan k if k < 0. The equation y = kx + b cannot represent a line which is parallel to the y-axis. The angle of inclination of such a line is right. Therefore in any case the angle of inclination satis es the condition 0 6 < .

We note also that any straight line that is not parallel to the y-axis is de ned by a single equation in the slope form.

Now we give conditions when the equations in the slope form de ne parallel, orthogonal or equal lines. We rewrite equations y = kx + b in the form kx + ( 1)y + b = 0 and apply conditions given above. For the straight lines `1 : y = k1x + b1 and `2 : y = k2x + b2 we obtain:

1.2.5Pencils of straight lines

Definition. The set of all straight lines passing through a xed point on the plane is called the pencil of lines. The common point of all straight lines is called the center of the pencil.

Let us nd equations of the lines from the pencil with the center at a point M0(x0; y0).

We assume rst that the line de ned by the equation Ax + By + C = 0 (jAj + jBj 6= 0) passes through the point M0. Then Ax0 + B0y + C = 0 and therefore C = Ax0 B0y. Substituting this value in the equation of the line we get:

Ax + By Ax0 By0 = 0;

Fig. 19. Pencil of straight lines

Any straight line passing through the point M0 is uniquely de ned by its normal vector. Equation (10) de nes such a line for an arbitrary nonzero vector ~n = fA; Bg. Therefore this equation de nes all straight lines from the pencil.

28 Chapter 1. Straight lines on the plane

Now we consider two lines

`1 : A1x + B1y + C1 = 0; `2 : A2x + B2y + C2 = 0

assuming that they are not parallel. Let M0 be the single intersection point of these lines.

Under these assumptions and in these notation, the following statement is true.

Theorem 3. The straight line belongs to the pencil with the center M0 if and only if it may be de ned by the equation

for some 1, 2 such that j 1j + j 2j 6= 0.

Proof. We assume rst that j 1j + j 2j 6= 0 and prove that (11) is the equation of a straight line passing through the point M0.

After simple transformations we rewrite (11) as follows:

( 1A1 + 2A2)x + ( 1B1 + 2B2)y + ( 1C1 + 2C2) = 0:

It is necessary to prove that the coe cients

1A1 + 2A2; 1B1 + 2B2

do not vanish at the same time. Assuming the contrary we get that

1A1 + 2A2 = 0; 1B1 + 2B2 = 0:

it means that the straight lines `1 and `2 are parallel and we get the contradiction. The case 2 6= 0 is reduced to a contradiction similarly.

Let us check that in the case j 1j + j 2j 6= 0 the straight line (11) passes through the point M0. Denoting M0(x0; y0) we obtain

A1x0 + B1y0 + C1 = 0; A2x0 + B2y0 + C2 = 0;

therefore

1(A1x0 + B1y0 + C1) + 2(A2x0 + B2y0 + C2) = 0:

Now we are going to prove that any straight line ` passing through the point M0 is be de ned by the equation (11) with some values 1 and 2. Such a straight line is uniquely de ned by any its point other than M0. Let M1(x1; y1) be this point. We substitute its coordinates into (11) with

The values A1x1 + B1y1 + C1 and A2x1 + B2y1 + C2 do not vanish at the same time since otherwise M1 would be the point on the both lines `1 and `2 and therefore M1 = M0. Let

1 = A2x1 + B2y1 + C2; 2 = (A1x1 + B1y1 + C1):

Then j 1j + j 2j 6= 0 and these values satisfy the equation (12). Therefore the equation (11) with the given values 1 and 2 de nes the line passing through the chosen point M1.

1.2.6Normal equations of straight lines

Definition. The normal equation of a straight line is the equation of

the form x cos + y sin p = 0 where p > 0.

Therefore for C = 0 we can take any sign, and in the case C 6= 0 we have to take the sign opposite to the sign of C. After division the equation of the line takes the form

We recall that the system of equations cos = a, sin = b provided that a2 + b2 = 1 has the single solution up to a term that is a multiple of 2 . In this case, if we require additionally that 0 6 < 2 then = arccos a if b > 0 and = 2 arccos a if b < 0.

In the case under consideration we have

It means that in the case C 6= 0, the equations (13) have the single solution such that 0 6 < 2 and have two such solutions if C = 0 since then the both signs are suitable.

Now we nd the geometric meaning of the parameters in the normal equation of the straight line. The value p is the distance of the origin from the line de ned by the equation x cos + y sin p = 0, since

j0 cos + 0 sin pj

p = p: cos2 + sin2

Here we took into account that p > 0 and cos2 + sin2 = 1.

Before discussing the meaning of the angle , we recall the fact from the school math course. For R > 0 the points M(R cos ; R sin ), 0 6 < 2