find the coordinates of a point on a straight line given the coordinates of the endpoints and the greater difference in coordinate?

Question

I have the coordinates of the endpoints of the line as $(x_1,y_1)$ and $(x_3,y_3)$. The difference in coordinate 'delta' refers to either $\Delta x$ or $\Delta y$ (We do not know which one it is) with respect to the first point $(x_1,y_1)$. However it is known that it is the larger difference i.e., Given $a>b$, if $x_2 = x_1+a$ and $y_2 = y_1+b$, then the 'delta' is $\Delta x$. Conversely, if $a<b$ and the other conditions remain the same, then the 'delta' is $\Delta y$. Please note that this is not the same as distance. Only the larger difference in coordinate is available.

For example, Given the straight line with coordinates $(0,0)$ and $(9,4)$ with the larger 'delta' as '3', we need to find the point $(x_2,y_2)$ with this delta.

Answer 1

Hint:

Note that the slope of the line is $$ m=\frac{y_3-y_1}{x_3-x_1} $$

and, for a point $P=(x_2,y_2)=(x_1+\Delta x,y_1+\Delta y)$ on this line we have: $$ m>1 \quad \rightarrow \quad \Delta x >\Delta y $$ $$ m<1 \quad \rightarrow \quad \Delta y >\Delta x $$

and for $m=1$ the two delta are the same. Can you see why? And can you do from this?