Question

If the lines $p_{1} x+ q_{1} y=1, p_{2} x+ q_{2} y=1$ and $p_{3} x +q_{3} y=1$ be concurrent, then the points $\left(p_{1}, q_{1}\right),\left(p_{2}, q_{2}\right)$ and $\left(p_{3}, q_{3}\right)$,

• A. are collinear
• B. form an equilateral triangle
• C. form a scalene triangle
• D. form a right angled triangle

Answer 1

Answer: (A)

$p _{1} x + q _{1} y =1, p _{2} x + q _{2} y =1 p _{3} x + q _{3} y =1$
Given lines are concurrent
$\Rightarrow \begin{vmatrix} p_{1} & q_{1} & 1 \\ p_{2} & q_{2} & 1 \\ p_{3} & q_{3} & 1 \end{vmatrix}=0$
$\Rightarrow p _{1}\left( q _{2}- q _{3}\right)- q _{1}\left( p _{2}- p _{3}\right)+\left( p _{2} q _{3}- p _{3} q _{2}\right)=0$
$\Rightarrow \left( p _{1} q _{2}- p _{2} q _{1}\right)+\left( p _{2} q _{3}- p _{3} q _{2}\right)+\left( p _{3} q _{1}- p _{1} q _{3}\right)=0$
The left hand side of the above equation is also equal to twice the area of a triangle with coordinates
$\left(p_{1}, q_{1}\right),\left(p_{2}, q_{2}\right),\left(p_{3}, q_{3}\right)$
Since it is equal to zero, $\left(p_{1}, q_{1}\right),\left(p_{2}, q_{2}\right),\left(p_{3}, q_{3}\right)$ are collinear.