How to find the locus from a general point

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Given a general point on a line, say $\left(\frac35x \ ; {2 \over 5}(x+3)\right)$ or any other general point, how can you find the locus as this point moves?

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Hint: Set $ b= \frac 25 (x+3), \, a = \frac 35 x$ and try to eliminate $x$. This will lead us to an equation between $a,b$. This equation reveals the locus of the points of the form $(a,b)$.