Question

The equation of one of the lines parallel to $4x-3y=5$ and at a unit distance from the point $(-1,-4)$ is

• A. $3x+4y-3=0$
• B. $3x+4y+3=0$
• C. $4x-3y+3=0$
• D. $4x-3y-3=0$

Answer 1

Answer: (D)

Equation of any lines parallel to
$4x-3y=5$ is $4x-3y+\lambda =0\dots \left(1\right)$
Since distance between the lines is one unit.
$\therefore$ Length of the perpendicular from $\left(-1,-4\right)$ to $4x-3y+\lambda =0$ is $1$.
$\Rightarrow \left|\frac{4\left(-1\right)-3\left(-4\right)+\lambda }{\sqrt{4^2+3^2}}\right|=1$
$\Rightarrow \left|8+\lambda \right|=5$
$\Rightarrow 8+\lambda =5$ or $8+\lambda =-5$
$\Rightarrow \lambda =-3$, or $\lambda =-13$
$\Rightarrow 4x-3y-3=0$ or $4x-3y-13=0$