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x^2 + xy - Ky^2=0 , the angle made by the line pairs of this equation is 45 degree. I got the value of k as 0 and -2, do we need to reject k=0 and conclude our answer as k=-2 ?

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asked by clive
  1. x^2 + xy - Ky^2=0

    To have this equation represented by two straight lines, we must have it in a factored form such as
    (x + ay)(x - by) = 0
    so that a+b = 1 and ab=-k

    e.g. if we had
    x^2 + xy - 6y^2 = 0
    (x+3y)(x-2y) = 0
    our two lines would be
    x+3y = 0 or x-2y = 0
    slope of first = -1/3
    slope of 2nd = 1/2
    for acute angle formed:
    using the formula
    tanθ = (m2 - m1)/(1 + m2m1)

    tanθ = (1/2 + 1/3)/(1 + (1/2)(-1/3)
    = (5/6) / (5/6)
    = 1
    θ = 45°
    What a lucky guess!

    check this graph:
    https://www.wolframalpha.com/input/?i=plot+x%5E2+%2B+xy+-+6y%5E2+%3D+0

    for yours:
    x^2 + xy - 2y^2 = 0
    (x+2y)(x-y) = 0
    slope of first = -1/2
    slope of 2nd = 1
    tanθ = (1 + 1/2)/(1 + (1)(-1/2)
    = (3/2)/(1/2)
    = 3
    θ = 71.56 degrees, not 45

    see: https://www.wolframalpha.com/input/?i=plot+x%5E2+%2B+xy+-+2y%5E2+%3D+0

    So for x^2 + xy - ky^2 = 0
    we need
    (x+ay)(x+by) = 0
    where a+b=1 and ab = -k
    also the two slopes are -1/a and -1/b
    tanθ = (-1/a + 1/b)/(1 + (-1/a)(-1/b))
    = (a-b)/(ab) / (1 + 1/ab)
    = (a-b)/(ab) / (ab + 1)(ab)
    = (a-b)/(ab+1)
    but tan45 = 1
    (a-b)/(ab+1) = 1
    a-b = ab + 1 , but we know that a+b=1 or
    b = 1-a

    a-(1-a) = a(1-a) + 1
    a-1+a = a-a^2 +1
    a^2 + a - 2 = 0
    (a-2)(a+1) = 0
    a = 2 or a = -1 , then
    b = -1 or b = 2

    so we could have
    (x+2y)(x-y) = 0 ---> x^2 + xy - 2y^2) = 0 , OUR CASE
    OR
    (x+y)(x-2y) = 0 ----> x^2 - xy - 2y^2 = 0, NOT OUR CASE

    k = ab = -2
    so k = -2 for each case

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    posted by Reiny
  2. I must have made some kind of error in this big mess.
    I can't reconcile the fact that k = 6 will produce the 45 degree angle, but both of our solutions somehow showed that k = 2

    I will look at it again later on today

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    posted by Reiny
  3. But I did it comparing with ax^2 + 2hxy + by^2=0 and got -2

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    posted by clive

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