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Thread: parabola equation

  1. #1
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    parabola equation

    Sir,

    Whether

    ax2 + by2 +cxy + dx + ey + f = 0

    could represent a parabola.

    If so,

    whether I can find vertex, focus, directrix, axis of parabola, length of latus rectum, etc.

    How to find the same.

    guidance please.

    with warm regards,

    Aranga
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  2. #2
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    Re: parabola equation

    Are there any restrictions on a, b, c, d, e or f?

    $ax^2+by^2+cxy+dx+ey+f=0$ is the general equation of a parabola (using your variables), $y=ax^2+dx+f$, when $b=c=0$ and $e=-1$.

    I just answered whether your equation "could" represent a parabola. Do you wish to clarify or add to your question?

    Best,
    Andy
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  3. #3
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    Re: parabola equation

    Thanks for the quick response.

    The Normal equation used in the book was

    ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 where h^2 = ab.

    Whether for such equation of parabola. Is it possible to find vertex, focus, axis of parabola, equation of directrix, length of latus rectum, etc.

    this is the actual question before me.

    kindly guide me.

    If possible with example.
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  4. #4
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    Re: parabola equation

    That is the general equation of a conic section. Certainly a parabola is a conic section but so is a circle, an ellipse, and a hyperbola. For certain values of a, b, f, g, h it could be any of those. In particular, if $h^2= ab$ then $ax^2+ by^2+ 2hxy= ax^2+ 2\sqrt{ab}xy+ by^2= (\sqrt{a}x+ \sqrt{b}y)^2$ so that equation becomes $(\sqrt{a}x+ \sqrt{b}y)^2+ dx+ ey+ f= 0$.

    Now, let $u= \sqrt{a}x+ \sqrt{b}y$ and $v= -dx- ey$ so we have $u^2- v+ f= 0$ or $v= u^2+ f$, a parabola. What are the vertex. focus, etc for that? What are they in terms of a, b, h, etc.?
    Thanks from abender and arangu1508
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  5. #5
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    Re: parabola equation

    Thank you very much. It is very useful. Now I got the idea to solve such problem in parabola. One more request. Any problem in this model so that I can do it myself and find vertex, focus, etc.

    with best wishes.
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