Question

for what value of A and B the following pair of equation will have infinitely many solutions x+2y=1 ; (a-b)x+(a+b)y=a+b-2

Answer 1

L₁ ≡ x + 2y = 1 ⇒ x + 2y -1 =0
L₂ ≡ (a-b)x + (a+b) y = a+ b-2 ⇒ (a-b)x + (a+b)y - (a+b-2) = 0

For two lines to have infinite solutions , they would be coincident lines

⇒ ratio of coefficients of x = ratio of coefficients of y = ratio of constants

⇒ 1/ (a-b) = 2/(a+b) = -1/ - (a+b-2)

fetches 2 equations

a+b = 2a - 2b  ⇒ 3b = a

and  a+b-2 = a-b ⇒ 2b = 2 ⇒ b = 1

so a = 3

Ans a =3 b = 1

 Hope this assists.