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Math Help - Product & Sum of Roots

  1. #1
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    Product & Sum of Roots



    (a) State the formula for determining the angle between the intersection of two lines.

    (b) Show that the angle bisector line: y = Mx between the lines y = m1*x and y = m2*x is given by (M - m1)(1 + M*m2) = (m2 - M)(1 + M*m1).

    (c) Hence show that the gradients of the two angle bisector lines is given by the quadratic in M:

    (m1 + m2)M^2 + 2(1-m1*m2)M - (m1 + m2) = 0.

    (d) State the product of the roots of this equation. Hence deduce the relationship between these two angle bisectors.

    N.B. the 1 and 2 in m1 and m2 are the bases of m (like in logs).

    Could someone please please show me how to do this question?
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  2. #2
    Senior Member Twig's Avatar
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    Gothenburg
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    hi

    Hi!

    I only did the first, maybe this will give you a hint how to proceed.
    Let the angle between the positive x-axis and y=m1x be called \theta
    and let the angle between y=mx and the positive x-axis be called \alpha

    Then the angle \beta is given by \theta - \alpha

    How do you figure out \theta \, \mbox{and } \alpha ?

    It has something to do with Tangens.
    Letīs say you have the line  y = x
    Then \frac{\Delta y}{\Delta x} = \mbox{ the angle between the positive x-axis and the line} where the index 1 and 2 stand for the different lines.
    Thus  tan(\frac{\Delta y}{\Delta x}) = \theta
    And  \theta = arctan(\frac{\Delta y}{\Delta x})

    Then you do the same for the other line.
    And you end up with:
     \beta = arctan(\frac{\Delta y_{2}}{\Delta x_{2}}) - arctan(\frac{\Delta y_{1}}{\Delta x_{1}})
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  3. #3
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    Exclamation

    I can't seem to be able to solve (b), (c) or (d). I have no clue how the equations given can be used to find the stuff. Could someone please give me a hint???
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