Definition. A line pencil is the set of straight line passing through a given point.
Arbitrarily varying the coefficient k in the equation of a line passing through a point Ì0(õ0,ó0), we obtain equations of all straight lines passing through the point with coordinates õ0, ó0.
All these straight lines constitute the line pencil centered at Ì0.
A line pencil can also be specified by two equations of lines from this pencil (which determine of these lines) the common point Ì0(õ0, ó0) by the coordinates of its center.
Suppose given intersecting straight lines with general equations
Consider the equations
where q1 and q2 undetermined multipliers, not both zero.
Let us prove that, for any q1 and q2, the straight line (13) passes through the point Ì0 (õ0; ó0), which is the intersection point of the two given lines.
The coordinates of the point Ì0 satisfy both equations
Consequently, the coordinates of Ì0satisfy equation (13), i.e.,
Thus, the point Ì0 belongs to the straight line (13) for any q1 and q2; but for particular values of q1 and q2, we obtain some straight line from the pencil.
In practice, it is more convenient to write an equation of a pencil with one parameter l:
Example. Draw a straight line through the intersection point of the lines õ+ó–3=0 and 2õ–ó+5=0 so that it passes through the point Ì(3;5).
Let us write the equation of the pencil of lines according to the formula. To find the required line, we must find the value of the parameter l:
We substitute the coordinates of the point Ì(3;5) into the equation of the pencil:
Substituting l in to the equation of the pencil, we obtain the required equation:
The Normal Equation of a Straight Line
The position of a straight line in the plane is completely determined by the distance from the line to the origin of coordinates, i.e., the length of the perpendicular from the origin to the fight line, and the unit normal vector
Let us construct an equation of a straight line from the length of the perpendicular ð and its angle a with the x-axis.
To this end we take an arbitrary point Ì on the line and find the projection of the vector on the unit normal vector .
The point Ì belongs to the given line if and only if
In terms of inner product of vectors, we have
Considering this product, we obtain
Equation (14) is the normal equation of the line in vector form. To pass to coordinates, note that the projections of the unit normal vector are
where a is the angle between this vector and the axis ÎX. Let the projections of the radius-vectors on the axes ÎÕ and ÎÓ are õ and ó, respectively. The inner product formula (14) implies
This equation is called the normal equation of the straight line. It is clear from equation (15) that the “normality” conditions are
(1) (the sum of squared coefficients of õ and ó equals 1);
(2) –ð<0 (the free element is negative).
Example. Which of the following equations are normal?
(a) , , this is not a normal equation;
(b) , and –4<0, this is a normal equation;
(c) , but 2>0, this is not a normal equation.
A normalizing factor. Suppose given a general equation of a straight line given:
Definition. The normalizing factor is the number such that the equation multiplied by this number is normal.
Let us multiply the general equation of a straight line by number m:
By definition, if this equation is normal, then the two normality conditions hold:
Removing parentheses, we obtain
extracting the root, we see that the normalizing factor is
the sign of the fraction is opposite to that of the free element of the general equation.
Example. Reduce the equation 8õ–6ó+5=0 to the normal form.
We find the normalizing factor
and multiply the equation by this factor term by term: