Question

The equations of three lines, AB, CD and EF are, $(b-c)x+(c-a)y+(a-b)=0$, $(c-a)x+(a-b)y+(b-c)=0$ and $(a-b)x+(b-c)y+(c-a)=0$. which one of the following inference is correct.

• A. The lines are parallel to each other
• B. AB and CD are perpendicular to EF
• C. All the lines are coincident
• D. The lines are concurrent

Answer 1

Answer: (D)

The lines are concurrent

As we can see in all equation,a term for all $+a,+b$ and $+c$ is present with a negative sign too.

$i.e.,-a,-b,-c$ but $x$ or $y$ is multiplied with some terms.

Hence if we put $x=y=1$ in all equations,all of them will be satisfied(i.e,=0).

$(b-c)x+(c-a)y+(a-b)=0$

$x=y=1$

$⟹b-c+c-a+a-b=0$

Similarily for other two equtions.

All lines are hence,concurrent at $x=y=1$

$\therefore D)Ans.$