# find equation of straight line through point A(-2,-7) and having intercepts of length 3 between straight lines 4x+3y=12 and 4x+3y=3

*x*+ 3

*y*= 12 and 4

*x*

*+*3

*y*= 3 are parallel lines

Let us first find the distance between the parallel lines

$\frac{3\times 4-3}{\sqrt{{4}^{2}+{3}^{2}}}=\frac{9}{5}=1.8$

Let

*A*be the angle made by the line with 4

*x*+ 3

*y*= 3

$\mathrm{tan}A=\frac{\sqrt{{3}^{2}-1.{8}^{2}}}{1.8}=\frac{4}{3}$

Let

*m*be the slope of the line

$\mathrm{tan}A=\frac{m-{\displaystyle \frac{4}{3}}}{1+{\displaystyle \frac{4m}{3}}}=\frac{4}{3}\Rightarrow 9m-12=12+16m\Rightarrow m=-\frac{24}{7}$

Equation of the line:

$y-\left(-7\right)=-\frac{24}{7}\left(x-\left(-2\right)\right)$

7

*y*+ 24

*x*+ 97=0

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