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What is the Homogeneous Equation of a Pair of Lines?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
A homogeneous equation of a pair of lines is a special type of quadratic equation in two variables (x and y) where every term has the same degree. It always represents two straight lines passing through the origin (0,0) on a graph.
Simple Example
Quick Example
Imagine you have two friends, Rahul and Priya, cycling from the same starting point (the origin). Rahul cycles along the path y = 2x, and Priya cycles along y = -x. If you combine these paths into one equation, it could look like 2x^2 + xy - y^2 = 0. This combined equation is a homogeneous equation because every term (2x^2, xy, -y^2) has a degree of 2.
Worked Example
Step-by-Step
Let's find the separate equations of lines represented by the homogeneous equation 2x^2 + 5xy + 2y^2 = 0.
STEP 1: Divide the entire equation by y^2 (assuming y is not zero) to get a quadratic equation in x/y.
(2x^2 / y^2) + (5xy / y^2) + (2y^2 / y^2) = 0
2(x/y)^2 + 5(x/y) + 2 = 0
---STEP 2: Let m = x/y. Substitute m into the equation.
2m^2 + 5m + 2 = 0
---STEP 3: Solve this quadratic equation for m using factorization.
2m^2 + 4m + m + 2 = 0
2m(m + 2) + 1(m + 2) = 0
(2m + 1)(m + 2) = 0
---STEP 4: Find the values of m.
2m + 1 = 0 => m = -1/2
m + 2 = 0 => m = -2
---STEP 5: Substitute back m = x/y to get the equations of the lines.
For m = -1/2: x/y = -1/2 => 2x = -y => 2x + y = 0
For m = -2: x/y = -2 => x = -2y => x + 2y = 0
ANSWER: The two separate lines are 2x + y = 0 and x + 2y = 0.
Why It Matters
Understanding these equations helps engineers design structures and circuits, and is crucial in computer graphics for rendering 3D objects. In AI/ML, these concepts form the basis for understanding how data points are separated or grouped. Learning this can open doors to careers in game development, architectural design, and even space technology at ISRO!
Common Mistakes
MISTAKE: Assuming all quadratic equations in x and y represent a pair of lines. | CORRECTION: Only *homogeneous* quadratic equations (where every term has the same degree) represent a pair of lines passing through the origin.
MISTAKE: Forgetting that a homogeneous equation always represents lines passing through the origin. | CORRECTION: Always remember that the origin (0,0) is a point on both lines represented by a homogeneous equation.
MISTAKE: Incorrectly factoring the quadratic equation after substituting m = x/y or y/x. | CORRECTION: Double-check your factorization steps carefully to ensure you get the correct values for m.
Practice Questions
Try It Yourself
QUESTION: Which of these is a homogeneous equation? (A) 3x^2 + 2y = 0 (B) 4x^2 - 9y^2 = 0 (C) x^2 + y^2 + 5 = 0 | ANSWER: B
QUESTION: Find the separate equations of the lines represented by x^2 - 4xy + 3y^2 = 0. | ANSWER: x - y = 0 and x - 3y = 0
QUESTION: If the equation ax^2 + 2hxy + by^2 = 0 represents two lines, and one of the lines is x + y = 0, find the relation between a, h, and b. | ANSWER: a - 2h + b = 0
MCQ
Quick Quiz
What is the degree of each term in a homogeneous equation of a pair of lines?
1
2
3
Can be any degree
The Correct Answer Is:
B
In a homogeneous equation of a pair of lines, like ax^2 + 2hxy + by^2 = 0, every term (ax^2, 2hxy, by^2) has a degree of 2. This is the defining characteristic.
Real World Connection
In the Real World
In civil engineering, when designing the intersection of two roads or railway tracks, the paths can sometimes be modeled using equations similar to a pair of lines. While real-world roads aren't perfectly straight, the underlying math for understanding how two linear paths meet is foundational. Also, in computer graphics, when rendering objects, the lines that form the edges of shapes are often represented mathematically, and understanding how pairs of lines behave is key to making realistic visuals on your mobile screen.
Key Vocabulary
Key Terms
HOMOGENEOUS: having terms of the same degree throughout | DEGREE OF A TERM: the sum of the powers of the variables in that term | ORIGIN: the point (0,0) where the x-axis and y-axis intersect | QUADRATIC EQUATION: an equation where the highest power of the variable is 2
What's Next
What to Learn Next
Next, explore 'General Equation of a Pair of Lines' to understand how lines not passing through the origin are represented. This will build on your knowledge of homogeneous equations and show you how to shift these lines around on the graph.
