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Pair of straight lines | Two dimensions geometry | ‪@drcolleger‬

An Introduction:
A straight line, often known as a line, is a one-dimensional infinite geometry having no width but just length.
A pair of straight lines is represented by a second-degree equation in two variables under specific conditions.
A pair of straight lines can also be represented as a straight line by multiplying two linear equations in x and y.

When the product of two linear equations in x and y indicates a straight line is multiplied together, a pair of straight lines is generated.
Let , L1=0 and L2=0 be two straight line equations.
If P(x1,y1) is a point on L1, the equation L1=0 is satisfied. If P(x1,y1) is a point on L2=0, then the equation is satisfied.
P(x1,y1) satisfies the equation L1L2=0 if it is located on L1 or L2.
∴ L1L2=0 denotes the pair of straight lines L1=0 and L2=0, and L1L2=0 denotes the joint equation of L1=0 and L2=0.

ax2+2hxy+by2+2gx+2fy+c=0 when we extend the above equation. This is a non-homogeneous second degree equation in x and y.
If a,b,h are not all zero, the general equation of a second degree homogeneous equation in x and y is ax2+2hxy+by2=0.
ax2+2hxy+by2=0 ,a pair of straight lines that pass through the origin

If a,b,h are not all zero, the general equation of a second-degree non-homogeneous equation in x and y is ax2+2hxy+by2+2gx+2fy+c=0.

Formulas for a Pair of Straight Lines
The following is a list of pair of straight lines formulas:
1. ax2+2hxy+by2=0 is a second-degree homogeneous equation that depicts a pair of straight lines flowing through the origin.
As a result, if h2(greater than)ab, the two straight lines are distinct and real
The two straight lines are coincident if h2=ab
If h2(less than)ab, the two straight lines with the origin as the point of intersection are imaginary

Angle Formed by Two Straight Lines
Consider the equation
ax2+2hxy+by2=0 ……(1)
for a pair of straight lines passing through the origin.
Let the slopes of these two lines be m1 and m2. By y dividing (1) by x2 and substituting y/x=m, we get
bm2+2hm+a=0
The roots of this quadratic in m will be m1 and m2. As a result, m1+m2= –2h/b and m1m2=ab.
The angle formed by the two lines is θ
Then,
tanθ=∣m2–m1|/|1+m2m1∣
=∣√(m1+m2)2–4m1m2/√1+m2m1∣
=∣√(–2h/b)2–4a/b/|√1+ab|
tanθ=∣2√h2–ab/a+b∣

The following are the outcomes of a general second-degree equation: ax2+2hxy+by2+2gx+2fy+c=0, which represents a pair of straight lines.
1. If ax2+2hxy+by2+2gx+2fy+c=0 denotes a pair of straight lines, the sum of their slopes is –2h/b and their product is a/b.
2.If tanθ=0, two lines will be parallel or coincident. i.e. if h2–ab=0
3. If tanθ is not defined, a+b=0, two lines will be perpendicular.
4. If the coefficient of xy=0, i.e. if h=0, two lines will be equally inclined to the axes.
5. The angle formed by two straight lines is given by tanθ=|2√h2–ab|/|a+b|.

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