equation of a line parallel to a given ine at a constant disance?

Question

what is the equation of a line parallel to a given line say y=x at a constant disance of 1 unit from it? I guess there will be 2 equations,one above x axis and other below x axis

Answer 1

The distance of a point $P=(x_P,y_P)$ from a stright line of equation $ax+by+c=0$ is given by: $$ d=\dfrac{|ax_P+by_P+c|}{\sqrt{a^2+b^2}} $$ (see here)

so, a generic point $(x,y)$ at distance $1$ from the line is such that: $$ \dfrac{|ax+by+c|}{\sqrt{a^2+b^2}}=1 $$ and , separating the two case for the absolute value, you have the equations of two stright lines.