Straight lines and Family of Straight lines Introduction

Homogeneous equation

• A rational, integral, algebraic equation of the variables x and y is said to be homogeneous equation of n^th degree in x and y, when the sum of the indices of x and y in every term is the same and is equal to n. Thus,

is a homogeneous equation of n^th degree in x and y.

•   Combined equation of two lines through the origin as a homogeneous equation of the second degree

Let

y = m₁x …(1)

y = m₂x …(2)

be the lines through the origin.

The equations (1) and (2) can be written as

y - m₁ x = 0 and y - m₂x = 0

Their combined equation is

This is a homogeneous equation of second degree in x and y.

•   A homogeneous equation of n^th degree in x and y represents n straight lines through the origin.

Let

This equation of n^th degree in x and y is a homogeneous equation and can be written in the form

i.e. say

This equation of n^th degree in z has n roots say m₁, m₂..... m_n.

The given equation can be split into n equations viz

Each of these n equations represents straight lines through the origin. Hence the given equation represents n lines through the origin.

It is to be noted that all the n straight lines need not be real and distinct. Some of them may be coincident and some imaginary.

Remark:

If the homogeneous equation in x and y is capable of being factorized into linear factors, then the equation represents as many straight lines as there are linear factors.

• Every homogeneous equation of second degree in x and y represents a pair of lines through the origin.

be a homogeneous equation of the second degree in x and y.

Dividing (1) throughout by x² , we get

. … (2)

…. (3)

Let m₁ and m₂ be the roots of the equation (2), then we have

Equation (3) can be written as

or

Equation (1) represents the above two lines.