<s>[INST] Problems involving LOCUS Okay so I'm really having a hard time finding the locus of curves or point of intersection of curves.
The problem : I am able to get the geometrical condition in most of the questions but then I have no idea what to do to get the locus.
The question :A variable straight line drawn through the point of intersection of the straight line $x+2y-4=0$ and $2x+y-4=0$ meet the coordinate axes in $A$ and $B$.Then what is the locus of midpoint of $AB$ ?
My try : I found the equation of the variable straight line as $3x+3y-8=0$,and the required mid-point in the question to be $(4/3,4/3)$.How do I proceed from here ? 
 [/INST] Let $M(a,b)$ be a point on our locus and $C\left(\frac{4}{3},\frac{4}{3}\right)$.
Thus, $$m_{MC}=\frac{b-\frac{4}{3}}{a-\frac{4}{3}}$$ and we have an equation of $AM$:
$$y-\frac{4}{3}=\frac{b-\frac{4}{3}}{a-\frac{4}{3}}\left(x-\frac{4}{3}\right),$$
which for $x=0$ gives
$$y=\frac{4}{3}-\frac{4}{3}\cdot\frac{b-\frac{4}{3}}{a-\frac{4}{3}}$$ and it gives an equation of the locus:
$$\frac{\frac{4}{3}-\frac{4}{3}\cdot\frac{b-\frac{4}{3}}{a-\frac{4}{3}}+0}{2}=b$$ or
$$b=\frac{2a}{3a-2}$$ or
$$y=\frac{2x}{3x-2}.$$
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