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Math Help - Angle bisector of paired lines

  1. #1
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    Angle bisector of paired lines

    By considering the set of points P(x, y) that are equidistant from the two lines represented by the equation ax^2+2hxy+by^2=0 show that the equation of the pair of angle bisectors of these lines is
    hx^2+(b-a)xy-hy^2=0

    I know that the pair of lines that are bisectors are perpendicular to each other and all the lines intersect at the origin. I don't know what else, or how to begin.
    Thanks!
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  2. #2
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    The pair of line can be written as

    y - m_1x = 0 and y - m_2x = 0

    where m_1 + m_2 = -\frac{2h}{b} and m_1m_2 = \frac{a}{b}

    Equations of angle bisectors are given by

    \frac{y - m_1x}{\sqrt{1+m_1^2}} - \frac{y - m_2x}{\sqrt{1+m_2^2}}= 0

    \frac{y - m_1x}{\sqrt{1+m_1^2}} + \frac{y -  m_2x}{\sqrt{1+m_2^2}}= 0

    The combined equation of the angle bisectors is given by

    \frac{(y - m_1x)^2}{1+m_1^2} - \frac{(y -  m_2x)^2}{1+m_2^2}= 0

    Simplify and substitute the values of m1+m2 and m1m2.
    Last edited by sa-ri-ga-ma; May 18th 2010 at 03:49 AM.
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  3. #3
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    umm how did you get the two first expressions? i don't understand
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  4. #4
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    Quote Originally Posted by arze View Post
    umm how did you get the two first expressions? i don't understand
    Equation of a straight line passing through the origin is given by

    y = mx, or y - mx = 0. If m1 and m2 are the slopes of a pair of straight lines, the combined equation is given by

    (y - m_1x)(y-m_2x) = 0

    y^2 - (m_1+m_2)x + m_1m_2x^2 = 0

    Compare this equation with equation given in the problem. You can see that

    m_1 + m_2 = -\frac{2h}{b}

    and m_1m_2 = \frac{a}{b}
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  5. #5
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    oops, sorry i meant these:
    \frac{y - m_1x}{\sqrt{1+m_1^2}} - \frac{y - m_2x}{\sqrt{1+m_2^2}}= 0
    \frac{y - m_1x}{\sqrt{1+m_1^2}} + \frac{y -  m_2x}{\sqrt{1+m_2^2}}= 0
    thanks!
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  6. #6
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    Quote Originally Posted by arze View Post
    oops, sorry i meant these:

    thanks!
    The angle bisector is the locus of points which are equidistant from the two straight lines.

    The two equations are two angle bisectors obtained by finding the perpendicular distance from a point (x, y)
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