O(0,0), A(1,2), B(3,4) are the vertices of tr | vedclass

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(D) Let $O(0,0), A(1,2), B(3,4)$.$1$. Median from $O$ to $AB$:The midpoint $D$ of $AB$ is $(\frac{1+3}{2}, \frac{2+4}{2}) = (2,3)$.The equation of the

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$3x^2+xy-2y^2=0$

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(D) Let $O(0,0), A(1,2), B(3,4)$.$1$. Median from $O$ to $AB$:The midpoint $D$ of $AB$ is $(\frac{1+3}{2}, \frac{2+4}{2}) = (2,3)$.The equation of the median $OD$ passing through $(0,0)$ and $(2,3)$ is $y = \frac{3}{2}x$,which simplifies to $3x - 2y = 0$.$2$. Altitude from $O$ to $AB$:The slope of $AB$ is $m_{AB} = \frac{4-2}{3-1} = \frac{2}{2} = 1$.The slope of the altitude $OP$ is $m_{OP} = -\frac{1}{m_{AB}} = -1$.The equation of the altitude $OP$ passing through $(0,0)$ with slope $-1$ is $y = -x$,which simplifies to $x + y = 0$.$3$. Joint equation:The joint equation of the altitude and median is $(3x - 2y)(x + y) = 0$.Expanding this,we get $3x^2 + 3xy - 2xy - 2y^2 = 0$,which is $3x^2 + xy - 2y^2 = 0$.

$O(0,0), A(1,2), B(3,4)$ are the vertices of $\triangle OAB$. The joint equation of the altitude and median drawn from $O$ is

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