Question

for which values of lamda do the pair of linear lamdax+y=lamba^2 and x+lamday+1

Answer 1

Sol : If the a1x+b1Y+ c1 = 0,a2x+b2y+c2 = 0 system has unique solution then a1/a2 ≠ b1/b2

Given linear equations λx + y = λ2 and x + λy = 1 λ/ 1 ≠ 1 / λ ⇒ λ2 ≠ 1 ⇒ λ ≠ ±1 ∴ λ ≠ ±1 then it has unique solution.

If the a1x+b1Y+ c1 = 0,a2x+b2y+c2 = 0 system has infinitely many solutions then a1/a2 = b1/b2 = c1/c2. λ/ 1 = 1 / λ = λ2 / 1

∴ λ = ±1 then it has infinitely many solutions. If the a1x+b1Y+ c1 = 0,a2x+b2y+c2 = 0 system has no solution then a1/a2 = b1/b2 ≠ c1/c2. λ/ 1 = 1 / λ ≠ λ2 / 1 λ3 ≠ 1 ⇒ λ ≠ 1

∴ λ = ±1 then it has no solution.