Find equation of line given length of a segment and point

Question

can anyone please help me with the following. I need to find possible equations of a line whose length of a segment is 5 between the lines with the following equations: $$x+2y+1=0$$ $$x+2y-1=0$$ And it contains a point P(-5,4). So far I've written this: $$y-4 = k(x+5)$$ But I am stuck there, I've created a system of equations but messed something up and can't get a solution, so any help would be much appreciated, cheers :)

Answer 1

Note that the distance between the two given parallel lines is $d=\frac2{\sqrt5}$ and the slope normal to them is $m=2$. Let $m’$ be the slope of the unknown line and $\theta$ the angle between the given lines and the unknown line. Then, $\cos\theta=\frac d5=\frac2{5\sqrt5}$ and

$$\tan\theta = \pm \frac{11}2 = \frac{m-m’}{1+mm’}$$

which yields $m’ = -\frac7{24},\>-\frac34$. Use the point-slope formula below to obtain the equations of the lines

$$y-4=m’(x+5)$$