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Math Help - one-parameter family of conics

  1. #1
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    one-parameter family of conics

    Hi to everybody!!!!

    I have to find a solution to these problems, but i'm not able.
    The question are:

    1.What is the general formula for the one parameter family of conics through the points (1,1), (1,-1), (-1,-1), (-1,1).

    I hope someone can help me! Thank you very much.

    2. Suppose R=[u] and S=[s] are two points in the projective space neither lying an a given conic defined by the symmetric bilinear form B. Show that RS is tangent to the conic if and only if B(u,u)B(v,v)-(B(u,v))^2=0.

    Thanks!
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  2. #2
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    Quote Originally Posted by womaninmaths View Post
    Hi to everybody!!!!

    I have to find a solution to these problems, but i'm not able.
    The question are:

    1.What is the general formula for the one parameter family of conics through the points (1,1), (1,-1), (-1,-1), (-1,1).

    ...
    Guten Morgen,

    there are 3 (one-parameter) families which passes through the vertices of this square:

    1. ellipses
    2. hyperbolae opening to the right and the left
    3. hyperbolae opening up and down

    The center of all conics is the origin.

    to #1.: The general equation of an ellipse with center at (0, 0) and the axes a and b is:

    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ..........Plug in the coordinates of (-1, 1) and (1, 1):

    \frac{1}{a^2}+\frac{1}{b^2}=1 ~\implies~b^2 = \frac{a^2}{a^2-1} .......... Plug in this term for bē into the original equation of the ellipse:

    \frac{x^2}{a^2}+\frac{y^2}{\frac{a^2}{a^2-1}}=1 ~\implies~ \frac{x^2}{a^2}+\frac{(a^2-1) \cdot y^2}{a^2}=1

    Don't forget to calculate the domain of the family!

    To demonstrate the effect of this equation I've attached a drawing with 1 < a < 5


    To #2, #3: Use the same method.
    Attached Thumbnails Attached Thumbnails one-parameter family of conics-fam_ellips.gif  
    Last edited by earboth; January 21st 2008 at 04:33 AM.
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