Lines are drawn parallel to the line 4x - 3y + 2 = 0, at a distance...

The equation of a line parallel to the line $ax + by + c = 0$ is $ax + by + k = 0 .$

Thus, the equation of a line parallel to $4x - 3y + 2 = 0$ is $4x - 3y + c = 0 .$

The distance of a line $ax + by + c = 0$ from origin is $\dfrac{\left|c\right|}{\sqrt{{a}^{2} + {b}^{2} + {c}^{2}}} .$

Given the line $4x - 3y + c = 0$ is at a distance of $\dfrac{3}{5}$ units from origin

$⇒ \dfrac{\left|c\right|}{\sqrt{16 + 9}} = \dfrac{3}{5}$

$⇒ \dfrac{\left|c\right|}{5} = \dfrac{3}{5}$

$⇒ c = \pm 3 .$

Hence, the equation of the lines are $4x - 3y + 3 = 0 and$4x - 3y - 3 = 0 .$</p><p>Clearly point \left(- \dfrac{1}{4}, \dfrac{2}{3}\right)$ satisfy the given equation $4x - 3y + 3 = 0 .$