Consider the set of all lines
Question:

Consider the set of all lines $p x+q y+r=0$ such that $3 p+2 q+4 r=0$. Which one of the following statements is true ?

1. The lines are all parallel.

2.  Each line passes through the origin.

3. The lines are not concurrent

The lines are concurrent at the point

4. $\left(\frac{3}{4}, \frac{1}{2}\right)$

Correct Option: , 4

Solution:

Given set of lines $p x+q y+r=0$

given condition $3 p+2 q+4 r=0$

$\Rightarrow \frac{3}{4} p+\frac{1}{2} q+r=0$

$\Rightarrow$ All lines pass through a fixed point $\left(\frac{3}{4}, \frac{1}{2}\right)$.