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Math Help - Conic Sections?

  1. #1
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    Conic Sections?

    Show algebraically that the graph of an equation of the form:
    Ax2 + Dx + Ey + F = 0, A 0

    a) Is a parabola if E 0.
    b) Is a vertical line if E = 0 and D2 - 4AF = 0.
    c) Is two vertical lines if E = 0 and D2 - 4AF > 0.
    d) Contains no points if E = 0 and D2 - 4AF < 0.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by tle8807 View Post
    Show algebraically that the graph of an equation of the form:
    Ax2 + Dx + Ey + F = 0, A 0

    a) Is a parabola if E 0.
    if E \neq 0, then we can write the equation in the form of Y = - \, \frac{Ax^2 + Dx + F}{E} you're done. but if you want the form y=4p(x-h), just factor the numerator by completing the square.

    Quote Originally Posted by tle8807 View Post
    b) Is a vertical line if E = 0 and D2 - 4AF = 0.
    If E = 0, then you have the quadratic equation Ax^2 + Dx + F = 0
    note that if you use the quadratic formula in solving x, D^2 - 4AF is the determinant, and if it is zero, you will be left with constant \frac{-D}{2A}

    Quote Originally Posted by tle8807 View Post
    c) Is two vertical lines if E = 0 and D2 - 4AF > 0.
    same with b). but here, note that for a quadratic equation in one variable, if the discriminant is positive, then you have two solutions for the variable, hence, you have two vertical lines.


    Quote Originally Posted by tle8807 View Post
    d) Contains no points if E = 0 and D2 - 4AF < 0.
    same reasong with c) but this time, if the discrimiant is negative, the graph does not intersect the x axis (in 2-d), hence "no" points. (i have to include the quotation marks..)
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