There are several formulae and equations used for the interactions of straight lines. We will explore those below:
Consider the equation (ax+by+c)(px+qy+r)=0 which is of second degree equation in two variables x and y. The equations ax+by+c=0 and px+qy+r=0 are linear in x and y and so, either of them represents a straight line. Let P(x,y) lies on the locus of the equation (ax+by+c)(px+qy+r)=0, then
ax+by+c=0 or px+qy+r=0
So P lies either on ax+by+c=0 or px+qy+r=0 (or both).
Thus a second degree equation in two variables x and y represents a pair of straight lines.
iIf a,b,h are real numbers not all zero, then ax2+2hxy+by2=0 is called homogeneous equation of second degree in x and y and ax2+2hxy+by2+2gx+2fy+c=0 is called the general equation of second degree in x and y.
If this general equation of second degree contains a straight line, then this equation can be written as the product of two linear factors in x and y.
Thus, if the locus of a second degree equation in x and y contains a straight line, then the equation represents a pair of straight lines. If the locus of a second degree equation in two variables x and y is a pair of straight lines, then we can write it as (ax+by+c)(px+qy+r)=0 where ax+by+c and px+qy+r are linear in x and y.
Note:
(1) The equation ax2+2hxy+by2=0 represents a pair of straight lines if and only if h2≥ab.
(2) If h2=ab, then the lines represented by the equation of locus are coincident.
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