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Indentify the typer of conic section and its vertices&foci. Calc II

  1. May 7, 2012 #1
    1. The problem statement, all variables and given/known data

    Identify the type of conic section whose equation is given as y^2+2y=4x^2+3
    Also find its vertices and foci
    2. Relevant equations

    I believe when you have x&y ^2 in an equation it can be either an ellipse or a hyperbola but if the signs are opposite it is a hyperbola. vertices ± a from center, foci ± c from center.

    3. The attempt at a solution
    I begin by rewriting the equation y^2+2y=4x^2+3 as
    y^2+2y-4x^2=3 thus allowing me to recognize that the powers or x/y ^2 are opposite, and thus the type of conic section is a hyperbola.
    I complete the square for the y(s) of the equation which gives me (y+1)^2-1
    and I now rewrite the equation as (y+1)^2-4x^2=3+1, (y+1)^2-4x^2=4
    there is no way of completing the square for 4x^2 however since I know the equation is in the form (x-h)^2 I will rewrite -4x^2 as -4(x-0)^2
    I will rewrite the equation again as:
    (y+1)^2-4(x-0)^2=4 and again in the form of (y-k)^2/a^2-(x-h)^2/b^2=1
    as, (y+1)^2/4 - (x-0)^2/1 = 1
    thus I have a^2=4, b^2=1, and a=2, b=1
    I know have all the necessary things to solve for the question.
    Center = (h,k)= (0,-1)
    vertices are ± a from center on axis, a =2
    thus vertices are (0,-1 ± 2 )
    foci are ± c from center, c^2=a^2+b^2= 1+4= √5
    foci: (0, -1±√5)

    That is all my work/solutions for this problem.
    Any help with what I may have done wrong and how to learn how to correct is, or even just saying "correct" is appreciated thank you.
  2. jcsd
  3. May 7, 2012 #2


    User Avatar
    Science Advisor

    Yes, that is "completing the square" for [itex]x^2[/itex].

    It all looks correct to me.
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