Correct Answer - Option 2 : 8/81
GIVEN:
‘α’ and ‘β’ are the roots of expression x2 – 12x + 27 = 0
FORMULA USED:
(α – β)2 = (α + β)2 – 4αβ
(α2 – β2) = (α – β)(α + β)
CALCULATION:
Since ‘α’ and ‘β’ are the roots of expression x2 – 12x + 27 = 0.
α + β = -b/a = 12 ---- (1)
αβ = c/a = 27 ---- (2)
From (1) and (2):
(α – β)2 = (α + β)2 – 4αβ
⇒ (α – β)2 = 144 – 108 = 36
⇒ α – β = 6
Now,
1/β2 – 1/α2
⇒ (α2 – β2)/α2β2
⇒ [(α – β)(α + β)]/α2β2
⇒ [(6 × 12)/272]
⇒ 8/81
∴ The value of 1/β2 – 1/α2 is 8/81.