Question

If the foot of the perpendicular from the origin to a straight line is at the point $\left(3,-4\right)$. Then, the equation of the line is

• A. $3x–4y=25$
• B. $3x-4y+25=0$
• C. $4x+3y-25=0$
• D. $4x+3y+25=0$

Answer 1

The correct option is A

$3x–4y=25$

Explanation for the correct option:

Let $P$$\left(3,-4\right)$ be the foot of the perpendicular

We know that slope of line joining two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$is $\frac{y_2-y_1}{x_2-x_1}$

$\therefore$Slope of line $OP$ is $m_1=\frac{-4-0}{3-0}=\frac{-4}{3}$

Let slope of line passing through $P$ be $m_2$

We know that for perpendicular lines, product of slopes $=-1$ i.e. $m_1\times m_2=-1$

$$\begin{gathered} \Rightarrow \frac{-4}{3}\times m_2=-1 \\ \Rightarrow m_2=\frac{3}{4} \end{gathered}$$

We know that equation of line passing through point $\left(x_1,y_1\right)$ with slope $m$ is given by $\left(y-y_1\right)=m\left(x-x_1\right)$

Therefore equation of line passing through $P$$\left(3,-4\right)$ with slope $m_2$is $\left(y+4\right)=\frac{3}{4}\left(x-3\right)\Rightarrow 3x-4y=25$

Hence, the correct option is option (A) i.e. $3x–4y=25$