1.

Find the equation of the line midway between the parallel lines `9x+6y-7=0` and 3x+2y+6=0`

Answer» Converting each of the given equation to the form `y=mx+C`,
We get, `9x+6y-7=0 Rightarrow y=(-3)/(2)x+(7)/(6)....(i)`
`3x+2y+6=0 Rightarrow y=(-3)/(2)x-3....(ii)`
Clearly, the slope of each one of the given line is `(-3)/(2)`
Let the given lines by `y=mx+C_(1) and y=mx+C_(2)`.
Then, `m=(-3)/(2), C_(1)=(7)/(6) and C_(2)=-3`
Let the required line. Then, L is parallel to each one of (i) and (ii) and equidistant from each one of them.
`therefore "slope of L"=(-3)/(2)`
Let the equation of L be y`=(-3)/(2)x+C......(iii)`
Then, distance between (i) and (iii) must be equal to the distance between (ii) and (iii).
`therefore (|C_(1)-C|)/(sqrt(1+m^(2)))=(|C_(2)-C|)/(sqrt(1+m^(2)))=|C_(1)-C|=|C_(2)-C|`
`Rightarrow |(7)/(6)-C|=|-3-C| Rightarrow |(7)/(6)-C|=|3+c|`
`Rightarrow (7)/(6)-C=3+C Rightarrow 2C=|(-11)/(6)Rightarrow C=(-11)/(2)`
`therefore "equation of L is y"=(-3)/(2)x-(11)/(12),i.e. 18+12y+11=0`
Hence, the line `18x+12y+11=0` is midway between the parallel line `9x+6y-7=0 and 3x+2y+6=0`.


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