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Math Help - Uniqueness of a line

  1. #1
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    Question Uniqueness of a line

    The question is:

    (a,b) and (c,d) are two distinct points in R^2. Prove there exists a unique line passing through them.

    I proved that a line exists:
    y=[(d-b)/(c-a)]x + [b-(d-b)a/(c-a)]

    But how do I prove that it is unique? I know the technique, you say that there are two and prove that they are the same, but how do I go about it in this problem? Thank you for your help!
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  2. #2
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    Quote Originally Posted by sfitz View Post
    The question is:

    (a,b) and (c,d) are two distinct points in R^2. Prove there exists a unique line passing through them.

    I proved that a line exists:
    y=[(d-b)/(c-a)]x + [b-(d-b)a/(c-a)]

    But how do I prove that it is unique? I know the technique, you say that there are two and prove that they are the same, but how do I go about it in this problem? Thank you for your help!
    Assume that there are at least 2 lines:

    y = m_1 x + c_1 and y = m_2 x + c_2.

    Sub the given points into each:

    y = m_1 x + c_1:

    b = a m_1 + c_1 .... (1)
    d = c m_1 + c_1 .... (2)

    Solve simultaneously for m_1 and c_1.


    y = m_2 x + c_2:

    b = a m_2 + c_2 .... (3)
    d = c m_2 + c_2 .... (4)

    Solve simultaneously for m_2 and c_2.


    Therefore establish that m_1 = m_2 and c_1 = c_2.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Assume that there are at least 2 lines:

    y = m_1 x + c_1 and y = m_2 x + c_2.
    .
    This does not prove it. What happens if the lines are vertical? Then we cannot write y=mx+b. Instead we should use ax+by = c. This handles all possible lines.
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