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Thread: question on intersection

  1. #1
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    question on intersection

    Q) Find the intersection of the lines $ax+by+a^2= 0$ and $bx−ay+b^2=0$

    I have attached my working out but i could not find the point of intersection

    i used two different methods
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  2. #2
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    Re: question on intersection

    $\begin{pmatrix}a &b \\ b &-a\end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix}-a^2 \\ -b^2 \end{pmatrix}$

    $\begin{pmatrix}a &b &| &-a^2\\ b &-a &| &-b^2\end{pmatrix}$

    $\left(
    \begin{array}{ccc}
    1 & \frac{b}{a} & -a \\
    b & -a & -b^2 \\
    \end{array}
    \right)$

    $\left(
    \begin{array}{ccc}
    1 & \frac{b}{a} & -a \\
    0 & -\frac{b^2}{a}-a & a b-b^2 \\
    \end{array}
    \right)$

    $\left(
    \begin{array}{ccc}
    1 & \frac{b}{a} & -a \\
    0 & 1 & \frac{a b (b-a)}{a^2+b^2} \\
    \end{array}
    \right)$

    $\left(
    \begin{array}{ccc}
    1 & 0 & -\frac{a^3+b^3}{a^2+b^2} \\
    0 & 1 & \frac{a b (b-a)}{a^2+b^2} \\
    \end{array}
    \right)$

    $x = -\dfrac{a^3+b^3}{a^2+b^2}$

    $y = \dfrac{a b (b-a)}{a^2+b^2}$
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