A combined equation of the two lines

A combined equation of the two lines through the origin is a homogenous equation of second degree

Let the two lines pass through the origin be y = m₁x and y = m₂x.

i.e. y - m₁x = 0 and y - m₂x = 0

Their combined equation is

This is clearly a homogenous equation of the second degree in x and y.

… (i)

Divide equation (1) by x²,

This is a quadratic form in m. This has two roots (say) m₁ and m₂_.

To show that a homogenous equation of degree n in x and y represents n straight lines passing through the origin

Any homogenous equation of the n^t^h degree in x and y is

This is an n^th degree equation in

and so has n roots. Let the roots be m₁, m₂, ......... m_n so that the given equation reduces to

Hence, the given equation represents n straight lines.

All these straight lines pass through the origin.

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Subjects
English
Math
Geometry