What is the Condition for a Pair of Lines? | Simple Explanation for Class 10

S6-SA1-0279

What is the Condition for a General Second Degree Equation to Represent a Pair of Lines?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A general second-degree equation in x and y looks like ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. For this equation to represent a pair of straight lines, a special condition must be met. This condition ensures that the equation can be factored into two linear equations.

Simple Example

Quick Example

Imagine you have two separate straight roads on a map. If you combine their equations, you get a second-degree equation. The condition tells us if a given complex equation actually describes two such separate roads, or if it's something else like a circle or an ellipse. It's like checking if two different cricket scores, say 150 and 120, can be described by one combined 'team performance' number, or if they are just random numbers.

Worked Example

Step-by-Step

Let's check if the equation x^2 - 5xy + 6y^2 = 0 represents a pair of lines.

Step 1: Compare the given equation with the general form ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. Here, a=1, 2h=-5 (so h=-5/2), b=6, g=0, f=0, c=0.

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Step 2: Write down the determinant condition: abc + 2fgh - af^2 - bg^2 - ch^2 = 0.

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Step 3: Substitute the values into the condition:
(1)(6)(0) + 2(0)(0)(-5/2) - (1)(0)^2 - (6)(0)^2 - (0)(-5/2)^2 = 0

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Step 4: Calculate each term:
0 + 0 - 0 - 0 - 0 = 0

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Step 5: The result is 0 = 0, which is true.

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Answer: Yes, the equation x^2 - 5xy + 6y^2 = 0 represents a pair of straight lines.

Why It Matters

Understanding how equations represent shapes is crucial in fields like Engineering and Physics, where you design structures or predict movement. For example, in AI/ML, similar mathematical conditions help classify data points into different groups. Architects and city planners also use these concepts to design layouts and predict traffic flow.

Common Mistakes

MISTAKE: Forgetting the '2' in '2hxy', '2gx', and '2fy' when comparing coefficients. | CORRECTION: Always remember the '2's in the general form ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. So if you have '3xy', then 2h=3, meaning h=3/2, not h=3.

MISTAKE: Mixing up the variables in the determinant condition, for example, using 'afg' instead of 'af^2'. | CORRECTION: Memorize the determinant condition correctly: abc + 2fgh - af^2 - bg^2 - ch^2 = 0. Double-check each term before substituting values.

MISTAKE: Making calculation errors, especially with signs, when substituting values into the determinant. | CORRECTION: Write down each substitution clearly and calculate term by term. Pay extra attention to negative signs and squaring negative numbers.

Practice Questions

Try It Yourself

QUESTION: Does the equation 2x^2 + 3xy + y^2 = 0 represent a pair of straight lines? | ANSWER: Yes

QUESTION: For what value of k does the equation x^2 + 4xy + ky^2 = 0 represent a pair of straight lines? | ANSWER: k=4

QUESTION: Determine if the equation x^2 - xy - 6y^2 - x + 8y - 2 = 0 represents a pair of straight lines. | ANSWER: Yes (Hint: Calculate the determinant: (1)(-6)(-2) + 2(4)(-1/2)(-1/2) - (1)(4)^2 - (-6)(-1/2)^2 - (-2)(-1/2)^2 = 12 + 2 - 16 - (-6)(1/4) - (-2)(1/4) = 14 - 16 + 3/2 + 1/2 = -2 + 4/2 = -2 + 2 = 0)

MCQ

Quick Quiz

Which of the following conditions must be true for the equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 to represent a pair of straight lines?

abc + 2fgh - af^2 - bg^2 - ch^2 = 1

abc + 2fgh - af^2 - bg^2 - ch^2 = 0

a+b+c = 0

h^2 - ab > 0

The Correct Answer Is:

The correct condition for a general second-degree equation to represent a pair of lines is that the determinant (abc + 2fgh - af^2 - bg^2 - ch^2) must be equal to zero. Options A, C, and D are incorrect conditions for this specific concept.

Real World Connection

In the Real World

Imagine a drone delivering a package in a city like Bangalore. Its flight path can be modeled using equations. If a drone needs to navigate through two intersecting 'no-fly zones', understanding if their boundaries form a pair of lines (or a different shape) using this condition helps plan a safe route. This is similar to how navigation apps like Google Maps use mathematical models to plot routes.

Key Vocabulary

Key Terms

GENERAL SECOND-DEGREE EQUATION: An equation of the form ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 | PAIR OF LINES: Two distinct straight lines represented by a single combined equation | DETERMINANT: A specific mathematical value calculated from the coefficients of the equation, used to check conditions | COEFFICIENTS: The numerical values (a, b, c, f, g, h) multiplying the variables in an equation

What's Next

What to Learn Next

Great job understanding this! Next, you can explore how to find the separate equations of the lines once you know the general equation represents a pair of lines. This will help you fully 'break down' the complex equation into its simpler components.