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Pairs of Straight Lines: An Introduction

A straight line, often known as a line, is a one-dimensional infinite geometry having no width but just length.

A straight line, often known as a line, is a one-dimensional infinite geometry having no width but just length. There are an endless number of points in a straight line. A two-variable linear equation is used to represent a straight line. 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. The concept of a pair of straight lines is particularly useful in mathematics since it helps us to simplify our complex issues. We’ll discover how to make a pair of straight lines, a pair of straight line formulae, a general question about a pair of straight lines, and more in this post.

Overview of a Pair of Straight Lines

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.

We get an equation of the type 

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>ab, the two straight lines are distinct and real

  • The two straight lines are coincident if h2=ab

  • If h2<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∣

Pair of Straight Lines Condition

When dealing with difficulties based on a pair of straight lines, some key conditions or outcomes are particularly useful.

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 tanθ=0, two lines will be parallel or coincident. i.e. if h2–ab=0

2. If tanθ is not defined, a+b=0, two lines will be perpendicular.

3. If the coefficient of xy=0, i.e. if h=0, two lines will be equally inclined to the axes.

4. The angle formed by two straight lines is given by tanθ=|2√h2–ab|/|a+b|.

5. 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.

Conclusion

A straight line, often known as a line, is a one-dimensional infinite geometry having no width but just length. There are an endless number of points in a straight line. A two-variable linear equation is used to represent a straight line. A pair of straight lines is represented by a second-degree equation in two variables under specific conditions.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. ax2+2hxy+by2=0 is a second-degree homogeneous equation that depicts a pair of straight lines flowing through the origin.If tanθ=0, two lines will be parallel or coincident. i.e. if h2–ab=0.If tanθ is not defined, a+b=0, two lines will be perpendicular.

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Frequently asked questions

Get answers to the most common queries related to the JEE Examination Preparation.

How do you handle a problem involving a pair of straight lines?

Ans. The answer is ax2+ 2hxy + by...Read full

How do you calculate the angle that is formed by the intersection of two straight lines?

Ans.  If a pair of straight lines is represented by the equation ax2+2hxy+by2+2gx+2fy+c=0, then the angle formed by the pair of straight lines may...Read full

How do you break apart the equation for two parallel straight lines?

Ans. In order to decouple the equation of a pair of parallel straight lines, we can proceed in one of two ways, as shown below: ...Read full

How do I determine the slope of the two straight lines?

Ans. If a pair of straight lines is represented by the equation ax2+2hxy+by2+2gx+2fy+c=0, then the equation m1+m2=–2h/b, and m1m2=a/b, where m1 a...Read full

What are some ways to locate a pair of parallel straight lines?

Ans. Two parallel straight lines are produced when the product of two linear equations in which x and y each represe...Read full