Question

Find the equation of the line that is parallel to 4x-5y = 2 and whose distance from the
given line is 5.

Answer 1

Since, equation of parallel line to ax+by+c=0 has of the form ax+by+c'=0

Now, In this case equation of parallel line to 4x-5y-2=0 has the form 4x-5y+c'=0

We also know that distance between two parallel line is

$$d=\frac{|c-c'|}{\sqrt{a^2+b^2}}$$

Thus,

$$5=\frac{|-2-c'|}{\sqrt{4^2+5^2}}\\\implies 5\sqrt{41}=|c'+2|\\ \implies c'=\pm5\sqrt{41}-2$$

Thus, equation of parallel lines are

$$4x-5y\pm 5\sqrt{41}-2=0$$