Question

If the roots of the equations x2+3x+2=0 and x2−x+λ=0  are in the same ratio then the value of λ  is given by-​

Answer 1

x² + 3x + 2 = 0

x² + 2x + x + 2 = 0

x(x + 2) + 1(x+2) = 0

(x+1)(x+2) = 0

x = -1 or x = - 2

The roots of the equation x² + 3x + 2 = 0 are -1 and -2.

Ratio of roots of the equation is 1:2 or 2:1

Given that the ratio of roots of x² - x + λ = 0 and x² + 3x + 2 = 0 are same.

So, the roots of the the equation of x² - x + λ = 0 are in the ratio 1:2 or 2:1

Case 1

suppose if they are in the ration 1:2

Let one root of x² - x + λ = 0 be y and the other root be 2y.

If α and β are the roots a quadratic equation ax² + bx + c = 0 then,

sum of the roots, α + β = -b/a

product of the roots αβ = c/a

Here, roots of x² - x + λ = 0 are y and 2y

a = 1, b = -1 and c = λ

so,

y + 2y = 1

3y = 1

y = 1/3

The roots of x² - x + λ = 0 are 1/3 and 2/3.

Product of the roots of x² - x + λ = 0 = λ

⇒ 1/3 * 2/3 = λ

⇒ λ = 2/9

Case 2:

Even if the roots are in the ratio 2:1, the value of λ remains same.

λ = 2/9

I hope, this helps you.