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, show that the points $(p_1,q_1),(p_2,q_2)$ and $(p_3,q_3)$ are collinear.

• A. vertices of right angle triangle
• B. vertices of an equilateral triangle
• C. vertices of an isosceles triangle
• D. Collinear

Answer 1

The correct option is A vertices of right angle triangle

Given:

$p_{1}x+q_{1}y=1,p_{2}x+q_{2}y=1$ and $p_{3}x+q_{3}y=1$

The lines will be concurrent if

$\left|\begin{array}{ccc}{p}_1& q_1& -1\\ p_2& q_2& -1\\ p_3& q_3& -1\end{array}\right|=0$

Or

$\frac{1}{2}\left|\begin{array}{ccc}{p}_1& q_1& 1\\ p_2& q_2& 1\\ p_3& q_3& 1\end{array}\right|=0$

Or

$△=0$

i.e., area of a triangle formed by the points $(p_1,q_1)$, $(p_2,q_2)$, and $(p_3,q_3)$ is zero and as such the points are collinear.