Question

The two given quadratic equations x² − 6x + a = 0 and x² – cx + 6 = 0 have one root in common and the roots of the second and first equations are integers which are in the ratio 3 : 4 . Then the common root is
1. -2
2. 2
3. -4
4. 4

Answer 1

Correct Answer - Option 2 : 2

Calculation:

Let α and 4β be the roots of

⇒ x² − 6x + a = 0

Let α and 3β be the roots of

⇒ x² – cx + 6 = 0

∴ from the relation between roots and coefficients

⇒ α + 4β = 6 and α × 4β = a

⇒ α + 3β = c and α × 3β = 6

∴ αβ = 2

∴ a = 8

∴ the first equation is x² −6x+ 8 =0

⇒ x² − 2x – 4x + 8 =0

⇒ x(x – 2) – 4(x – 2) =0

∴ x = 2, 4

For α = 4,

4β = 2

3β = 3/2 (not an integer)

∴ α = 2.

∴ the common root of both the equations is 2.