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Math Help - Find loci (parameter involved)

  1. #1
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    Find loci (parameter involved)

    Two straight line: kx+y=1-2k, x-k(y-2)-1. Let P be the point of intersection of these 2 lines. Find the equation of locus of P when the value of k varies.

    I found that k = (x-y)/(x+y), x= -(k+1)(2k-1)/(k^2+1), y= (k-1)(2k-1)/(k^2+1).
    But I can't find a good method to eliminate the parameter k in above two equations.
    Could any one help me?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Find loci (parameter involved)

    Quote Originally Posted by piscoau View Post
    Two straight line: kx+y=1-2k, x-k(y-2)-1. Let P be the point of intersection of these 2 lines. Find the equation of locus of P when the value of k varies.
    Supposing the lines are kx+y=1-2k and x-k(y-2)-1=0 then, k=\frac{1-y}{x+2}=\frac{1-x}{y-2}\Leftrightarrow \ldots \Leftrightarrow x^2-y^2+x+3y-4=0 (hyperbola).
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