Find the locus of the middle point of the intercept on the line y=x+c made by the lines 2x+3y=5 & 2x+3y=8, c being a parameter?

Question

Here's my shot: since the two lines are parallel, I figured that the middle point should be equidistant from the parallel lines,so using distance formula:-

$ \frac{2x+3y-5}{\sqrt{13}}=\frac{2x+3y-8}{\sqrt{13}}$

I get, $4x+6y-13=0$(considering one equation +ve and other -ve) which seems to be the right answer, but my question is, is there any other standard method to solve this question?

Answer 1

The locus traced by all the midpoints is just the line in the middle: $2x+3y=\frac{13}{2}$ or $4x+6y-13=0$.

Answer 2

You need to solve the following system: $$2x+3y=\frac{5+8}{2}$$ and $$y=x+c.$$