Where does this garbage line come from?

Question

A question reads, "Find equation of the line which is equidistant from parallel lines $9x + 6y – 7 = 0$ and $3x + 2y + 6 = 0$."

I solved it by doing the following:-

1. Finding the distance between the two given lines, halved it to find the distance between the required line which is $9x + 6y + C = 0$ and the line $3x + 2y + 6 =0$.

   2. Took a point $(-2,0)$ lying on the line $3x + 2y + 6 = 0$.

   3. Used the formula for perpendicular distance between a point and a line, but due to modulus, I get two solutions for $C$ (and for the equation).

The two solutions for $C$ are $11/2$ (which is correct) and the other garbage value $61/2$ which is obviously wrong as it would lie above both the given equations as its $y$-intercept is larger than both the equations.

My question is - Why does this garbage value arrive?

Thank you.

Answer 1

Visualise a set of railway tracks, its two metal lines being those whose equations you're given. In step 1, you worked out the track's half-width. In step 2, you marked a point on, say, the easternmost of the lines. In step 3, you moved perpendicular to that line through a half-width's worth. Going West takes you inside the track, halfway to the other line; going East juts out in the opposite direction.

Answer 2

If you have two equations of parallel lines of the form

$$ax + by = c_1\tag{a}$$ and $$ax + by = c_2\tag{b}$$ then it is easy to see that the equidistant line between them has equation $$ax+by = \frac{c_1+c_2}{2}\tag{c}$$ (so just take the average of the right hand sides).

Rewrite your equations as

$$9x+6y=7 \tag{1}$$ $$9x+6y=-18\tag{2}$$

and apply the above. Then rewrite it to your favourite form.