Find the equation from perpendicular and line equation

Question

Find the equation of the line which is at right angles to $3x + 4y =12$, such that its perpendicular distance from the origin is equal to the length of the perpendicular from $(3, 2)$ on the given line.​ How to solve this?

Answer 1

hint

The equation of the line $ P $ which is perpendicular to the line $ D $ whose equation is $$ax+by=c$$ will be $$P : bx-ay=d $$

The perpendicular distance from a point $ (\alpha,\beta) $ to the line $ D $ is

$$\frac{|a\alpha+b\beta-c|}{\sqrt{a^2+b^2}}.$$