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| Introduction |
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Homogeneous equation |
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| A rational, integral, algebraic equation of the variables x and y is said to be homogeneous equation of nth 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, |
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 |
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| is a homogeneous equation of nth degree in x and y. |
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Combined equation of two lines through the origin as a homogeneous equation of the second degree |
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| Let |
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| y = m1x …(1) |
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| y = m2x …(2) |
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| be the lines through the origin. |
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| The equations (1) and (2) can be written as |
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| y - m1 x = 0 and y - m2x = 0 |
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| Their combined equation is |
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 |
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| This is a homogeneous equation of second degree in x and y. |
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A homogeneous equation of nth degree in x and y represents n straight lines through the origin. |
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| Let |
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 |
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| This equation of nth degree in x and y is a homogeneous equation and can be written in the form |
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 |
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| i.e. say |
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 |
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| This equation of nth degree in z has n roots say m1, m2..... mn. |
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The given equation can be split into n equations viz |
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 |
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| Each of these n equations represents straight lines through the origin. Hence the given equation represents n lines through the origin. |
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| 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. |
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| Remark: |
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| 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. |
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Every homogeneous equation of second degree in x and y represents a pair of lines through the origin. |
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 |
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| be a homogeneous equation of the second degree in x and y. |
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| Dividing (1) throughout by x2 , we get |
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. … (2) |
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 |
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…. (3) |
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| Let m1 and m2 be the roots of the equation (2), then we have |
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 |
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| Equation (3) can be written as |
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| or |
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| Equation (1) represents the above two lines. |
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