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Thread: Analytic Geometry

  1. #1
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    Analytic Geometry

    If the intersect curve of $\displaystyle Ax + By + Cz + D = 0$ and $\displaystyle x^2 + y^2 - z^2 = 1$ are two lines

    show that: $\displaystyle A^2 + B^2 = C^2 + D^2$

    My idea is: joint the two equations: $\displaystyle \left\{\begin{matrix} Ax + By + Cz + D = 0 \\
    x^2 + y^2 - z^2 = 1 \end{matrix}\right.$

    then elimination one parameter: because $\displaystyle Ax + By + Cz + D = 0$ is a plane, so $\displaystyle A^2 + B^2 + C^2 \not = 0$

    so let's Assume $\displaystyle A\not = 0$, then $\displaystyle x = - \frac {1}{A}\left(By + Cz + D\right)$, take into $\displaystyle x^2 + y^2 - z^2 = 1$ in order to eliminate $\displaystyle x$, we

    get: $\displaystyle \left(1 + \frac {B^2}{A^2}\right)y^2 + \left(\frac {C^2}{A^2} - 1\right)z^2 + \frac {2BD}{A^2}y + \frac {2CD}{A^2}z + \frac {D^2}{A^2} - 1 = 0$

    we know: it is a plane curve, and we know ,it replaces two plane lines if and only if : $\displaystyle I_2 \leq 0, I_3 = 0$

    where: $\displaystyle I_3 = \begin{vmatrix} 1 + \frac {B^2}{A^2} & 0 & \frac {BD}{A^2} \\
    0 & \frac {C^2}{A^2} - 1 & \frac {CD}{A^2} \\
    \frac {BD}{A^2} & \frac {CD}{A^2} & \frac {D^2}{A^2} - 1 \end{vmatrix}$ $\displaystyle I_2 = \begin{vmatrix} 1 + \frac {B^2}{A^2} & 0 \\
    0 & \frac {C^2}{A^2} - 1 \end{vmatrix}$
    and I want to get the conclusion from : $\displaystyle I_2 \leq 0, I_3 = 0$, But I failed, Help me
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  2. #2
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
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    You have made a mistake here:
    $\displaystyle \left(1 + \frac {B^2}{A^2}\right)y^2 + \left(\frac {C^2}{A^2} - 1\right)z^2 + \frac {2BD}{A^2}y + \frac {2CD}{A^2}z + \frac {D^2}{A^2} - 1 = 0$

    It should be
    $\displaystyle \left(1 + \frac {B^2}{A^2}\right)y^2 + \left(\frac {C^2}{A^2} - 1\right)z^2 + \frac{2BC}{A^2}yz + \frac {2BD}{A^2}y + \frac {2CD}{A^2}z + \frac {D^2}{A^2} - 1 = 0$
    since the intersection are two lines, consider the above expression as a equation with y as unknown variable, simplify the equation by multiplying $\displaystyle A^2$, its discriminant
    $\displaystyle \triangle_{y}=(A^4+B^2A^2-C^2A^2)z^2-2A^2CDz+(A^4+B^2A^2-D^2A^2)$
    must be a perfect square expression containning z as variable. that is to say,
    $\displaystyle (A^2+B^2-C^2)z^2-2CDz+(A^2+B^2-D^2)$
    is a perfect square expression with respect to z. thus its discriminant
    $\displaystyle \triangle_{z}=4C^2D^2-4(A^2+B^2-C^2)(A^2+B^2-D^2)=0$
    which gives $\displaystyle A^2+B^2=C^2+D^2$.
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