Question

If the lines$x+ay+a=0,bx+y+b=0$ and $cx+cy+1=0$ are concurrent,then write the value of $2abc-ab-bc-ca.$

Answer 1

$x+ay+a=0$ ...(1)

$bx+y+b=0$ ...(2)

$cx+cy+1=0$ ...(3)

Solving (1) and (2) we get

$y=\frac{b-ab}{ab-1},x=\frac{a-ab}{ab-1}$

Put x and y in equation (3)

$c\left(\frac{a-ab}{ab-1}\right),+c\left(\frac{b-ab}{ab-a}\right)+1=0$

$ca-cab+cb-abc+ab-1=0$

$2abc-ab-bc-ca=-1$