Maharashtra Board Class 12 Maths Part 1 Chapter 4 Pair Of Straight Lines PDF Download

Pair Of Straight Lines

An equation which represents two lines is called the combined equation of those two lines. Let u = a₁x + b₁y + c₁ and v = a₂x + b₂y + c₂. Equation u = 0 and v = 0 represent lines. We know that equation u + kv = 0, k ∈ R represents a family of lines. Let us interpret the equation uv = 0.

Theorem 4.1

The equation uv = 0 represents the combined equation of lines u = 0 and v = 0

Proof

Consider the lines represented by u = 0 and v = 0

∴ a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.

Let P(x₁, y₁) be a point on the line u = 0.

∴ (x₁, y₁) satisfy the equation a₁x + b₁y + c₁ = 0

∴ a₁x₁ + b₁y₁ + c₁ = 0

To show that (x₁, y₁) satisfy the equation uv = 0.

(a₁x₁ + b₁y₁ + c₁) (a₂x₁ + b₂y₁ + c₂) = 0 (a₂x₁ + b₂y₁ + c₂) = 0

Therefore (x₁, y₁) satisfy the equation uv = 0.

This proves that every point on the line u = 0 satisfy the equation uv = 0.

Similarly we can prove that every point on the line v = 0 satisfies the equation uv = 0.

Now let R(x', y') be any point which satisfy the equation uv = 0.

∴ (a₁x' + b₁y' + c₁) (a₂x' + b₂y' + c₂) = 0

∴ (a₁x' + b₁y' + c₁) = 0 or (a₂x' + b₂y' + c₂) = 0

Therefore R(x', y') lies on the line u = 0 or v = 0.

Every points which satisfy the equation uv = 0 lies on the line u = 0 or the line v = 0.

Therefore equation uv = 0 represents the combined equation of lines u = 0 and v = 0.

Remark

1) The combined equation of a pair of lines is also called as the joint equation of a pair of lines.

2) Equations u = 0 and v = 0 are called separate equations of lines represented by uv = 0.

Teacher's Note

When two straight lines meet at a point, their equations can be written as one equation. For example, if you draw two roads crossing each other in your village, one equation can show both roads together.

Exam Trick

Remember: If u = 0 and v = 0 are two lines, then uv = 0 gives both lines together. It is like multiplying two things - if either one is zero, the whole product is zero.

Points to Remember

Combined equation of two lines means one equation that shows both lines.
If u = 0 and v = 0 are separate lines, then uv = 0 is the combined equation.
Every point on either line satisfies the combined equation uv = 0.
The combined equation is also called the joint equation.

Solved Examples

Ex. 1) Find the combined equation of lines x + y – 2 = 0 and 2x – y + 2 = 0

Solution: The combined equation of lines u = 0 and v = 0 is uv = 0

∴ The combined equation of lines x + y – 2 = 0 and 2x – y + 2 = 0 is (x + y – 2)(2x – y + 2) = 0

∴ x(2x – y + 2) + y(2x – y + 2) – 2(2x – y + 2) = 0

∴ 2x² – xy + 2x + 2xy – y² + 2y – 4x + 2y – 4 = 0

∴ 2x² + xy – y² – 2x + 4y – 4 = 0

Ex. 2) Find the combined equation of lines x – 2 = 0 and y + 2 = 0.

Solution: The combined equation of lines u = 0 and v = 0 is uv = 0.

∴ The combined equation of lines x – 2 = 0 and y + 2 = 0 is (x – 2) (y + 2) = 0

∴ xy + 2x – 2y – 4 = 0

Ex. 3) Find the combined equation of lines x – 2y = 0 and x + y = 0.

Solution: The combined equation of lines u = 0 and v = 0 is uv = 0.

∴ The combined equation of lines x – 2y = 0 and x + y = 0 is (x – 2y) (x + y) = 0

∴ x² – xy – 2y² = 0

Ex. 4) Find separate equation of lines represented by x² – y² + x + y = 0.

Solution: We factorize equation x² – y² + x + y = 0 as (x + y) (x – y) + (x + y) = 0

∴ (x + y) (x – y + 1) = 0

Required separate equations are x + y = 0 and x – y + 1 = 0.

Teacher's Note

Finding separate equations means breaking one equation into two simple equations. Like breaking \(2x + 2y = 4\) into two parts: \(x = 1\) and \(y = 1\).

Exam Trick

To find separate equations, try to factorize the combined equation like you factorize numbers. Look for common factors.

Points to Remember

Combined equation means two lines written as one equation.
Separate equations means the two simple line equations by themselves.
We can find separate equations by factorizing the combined equation.
Always check your answer by multiplying the separate equations.

Homogeneous Equation Of Degree Two

Degree Of A Term

Definition: The sum of the indices of all variables in a term is called the degree of the term. For example, in the expression x² + 3xy – 2y² + 5x + 2 the degree of the term x² is two, the degree of the term 3xy is two, the degree of the term –2y² is two, the degree of 5x is one. The degree of constant term 2 is zero. Degree of '0' is not defined.

Homogeneous Equation

Definition: An equation in which the degree of every term is same, is called a homogeneous equation.

For example: x² + 3xy = 0, 7xy – 2y² = 0, 5x² + 3xy – 2y² = 0 are homogeneous equations. But 3x² + 2xy + 2y² + 5x = 0 is not a homogeneous equation. Homogeneous equation of degree two in x and y has form ax² + 2hxy + by² = 0.

Theorem 4.2

The combined equation of a pair of lines passing through the origin is a homogeneous equation of degree two in x and y.

Proof

Let a₁x + b₁y = 0 and a₂x + b₂y = 0 be any two lines passing through the origin. Their combined equation is (a₁x + b₁y) (a₂x + b₂y) = 0

a₁a₂x² + a₁b₂xy + a₂b₁xy + b₁b₂y² = 0

(a₁a₂) x² + (a₁b₂ + a₂b₁) xy + (b₁b₂) y² = 0

In this if we put a₁a₂ = a, a₁b₂ + a₂b₁ = 2h, b₁b₂ = b, we get, ax² + 2hxy + by² = 0, which is a homogeneous equation of degree two in x and y.

Teacher's Note

Homogeneous means all terms have the same power. Like \(2x^2 + 3xy + 5y^2 = 0\) - all terms have power 2. Every line through the origin gives such equations.

Exam Trick

Check if all terms have the same degree. If yes, it is homogeneous. If there is a constant like \(+ 5\), then it is not homogeneous.

Points to Remember

Homogeneous equation has all terms with same degree.
Lines passing through origin give homogeneous equations.
Form: \(ax^2 + 2hxy + by^2 = 0\)
Count the power in each term to check if it is homogeneous.

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