Understanding the Equation of Pair of Straight Lines Passing through Origin and Perpendicular to Given Lines

Welcome to a comprehensive exploration on the equation of pair of straight lines passing through the origin and perpendicular to the lines represented by the given equation. This article delves into the mathematical intricacies involved in finding such an equation and aims to provide you with clear, step-by-step instructions to solve similar problems effectively.

The Given Equation and Its Factorization

Let's start with the given equation: 2x^2-5xy-2y^20. This equation represents a pair of straight lines passing through the origin. To understand the pair of lines it represents, we need to factorize this equation:

2x^2-5xy-2y^2 (x - y)(2x 5y) 0

This factorization reveals the two individual lines: x - y 0 and 2x 5y 0. These two lines intersect at the origin (0,0).

In fact, we can derive another form of this equation: 5x^2 7xy 2y^2 0. This transformation is achieved through an algebraic manipulation, which we will explore later.

Multiplying to Find the Perpendicular Lines

Now, let's find the equation of a line that is perpendicular to the given lines and also passes through the origin. The slope of a line perpendicular to another is the negative reciprocal of the original slope.

Given the equations x - y 0 and 2x 5y 0, the slopes of these lines are 1 and -5/2, respectively. Therefore, the slopes of the perpendicular lines are -1 and 2/5.

The equations of the lines with these slopes and passing through the origin are:

y -x and y frac{2}{5}x

These can be written as:

x y 0 and 5x - 2y 0

The joint equation of these two lines is:

x y 5x - 2y 0 Rightarrow 6x - y 0

However, we need to derive the combined equation that represents these two perpendicular lines. Let's proceed to the next section.

Deriving the Combined Equation

A more systematic approach involves using the auxiliary equation method. The auxiliary equation of the given pair of lines is:

5m^2 - 7m 2 0

By solving this quadratic equation, we find the slopes of the original lines:

m frac{1}{1} 1 and m frac{2}{5}

The slopes of the perpendicular lines are the negative reciprocals of these, which are -1 and -5/2, respectively. Therefore, the equations of the lines perpendicular to these are:

y -x and y -frac{5}{2}x

The joint equation of these two lines is:

x y 5x - 2y 0 Rightarrow -x - y 5x - 2y 0 Rightarrow 5x^2 7xy 2y^2 0

Alternative Method using Substitution

Another approach involves a 90-degree rotation of the axes. This is achieved by substituting x with -y and y with x in the original equation:

2x^2 - 5xy - 2y^2 0 Rightarrow 2(-y)^2 - 5(-y)x - 2x^2 0 Rightarrow 2y^2 5yx - 2x^2 0 Rightarrow 2x^2 7xy 5y^2 0

This approach also leads us to the same combined equation:5x^2 7xy 2y^2 0

Happy New Year!

As a special note, let’s celebrate the new year with a fun problem! The equation 2x^2 - 5xy - 2y^2 0 can be rewritten as two lines intersecting at the origin:

2x - yx - 2y 0

The lines are perpendicular to the lines given and passing through the origin are:

x - y 0 and 2x y 0

Multiplying these, we get:

x - y 0 and 2x y 0 Rightarrow x - y 2x y 0 Rightarrow 5x^2 7xy 2y^2 0

Conclusion

In conclusion, the equation of pair of straight lines passing through the origin and perpendicular to the lines represented by 2x^2 - 5xy - 2y^2 0 is 5x^2 7xy 2y^2 0. This result can be achieved through factorization, the auxiliary equation method, or a 90-degree axis rotation. Understanding these techniques is crucial for tackling similar problems in advanced mathematics.