Question
Consider the set of all lines $$px+qy+r=0$$ such that $$3p+2q+4r=0.$$ Which one of the following statements is true?
A.
The lines are concurrent at the point $$\left( {\frac{3}{4},\,\frac{1}{2}} \right).$$
B.
Each line passes through the origin.
C.
The lines are all parallel.
D.
The lines are not concurrent.
Answer :
The lines are concurrent at the point $$\left( {\frac{3}{4},\,\frac{1}{2}} \right).$$
Solution :
The given equations of the set of all lines
$$px+qy+r=0 .....(1)$$
and given condition is :
$$\eqalign{
& 3p + 2q + 4r = 0 \cr
& \Rightarrow \frac{3}{4}p + \frac{2}{4}q + r = 0.....(2) \cr} $$
From (1) & (2) we get :
$$\therefore x = \frac{3}{4},\,\,\,\,\,y = \frac{1}{2}$$
Hence the set of lines are concurrent and passing through the fixed point $$\left( {\frac{3}{4},\,\frac{1}{2}} \right).$$