Find the value of lamda from given equation of pair of straight lines

Question

If the two lines are represented by the the equation (lamda)x^2+2y^2–5xy+5x–7y+3=0,then lamda is equal to

Answer 1

Set x=0. Then your equation (λx^2+2y^2–5xy+5x–7y+3=0) becomes:

λ(0^2) + 2(y^2) – 5(0)(y) + 5(0) – 7y + 3 = 0

which is the same as

2(y^2) – 7y + 3 = 0

which factors to

(2y – 1) (y – 3) = 0

So, y = 3 or y = 0.5

Now set y=0. Then your equation (λx^2+2y^2–5xy+5x–7y+3=0) becomes:

λ(x^2) + 2(0^2) – 5(x)(0) + 5x – 7(0) + 3 = 0

which is the same as

λ(x^2) + 5x + 3 = 0

which factors to

either (λx + 1) (x + 3) = 0 or (λx + 3) (x + 1) = 0

So, λ equals 4/3 (in the first case) or λ equals 2 (in the second case)

If λ equals 4/3, then ((4/3)x + 1) (x + 3) = 0, so x = -3 or x = -3/4

If λ equals 2, then (2x + 3) (x + 1) = 0, so x = -3/2 or x = -1

So your options are y = 3 or y = 0.5; and x = -3, or -3/4, or -3/2, or -1.