Question

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Answer 1

Question:-

If α , β are the roots of 9x² + 6x + 1 = 0 , then the equation with roots 1/α , 1/β is

A) 2x² + 3x + 18 = 0.

B) x² + 6x - 9 = 0

C) x² + 6x + 9 = 0

D) 2x² + 3x + 9 = 0

Answer:-

Given:-

α , β are the roots of 9x² + 6x + 1 = 0.

On comparing it with standard form of a quadratic equation i.e., ax² + bx + c = 0 ;

Let,

• a = 9
• b = 6
• c = 1.

We know that,

Sum of the roots = - b/a

⟹ α + β = - 6/9

⟹ α + β = - 2/3 -- equation (1)

Product of the roots = c/a

⟹ αβ = 1/9 -- equation (2)

Now,

We have to find the quadratic equation whose roots are 1/α , 1/β.

We know,

General form of a quadratic equation : x² - (sum of the roots)x + Product of roots = 0

★ Sum of the roots = 1/α + 1/β

⟹ Sum of the roots = (β + α)/αβ

⟹ Sum of the roots = ( - 2/3) / (1/9) = ( - 2/3) × 9.

[ ∵ From equations (1) & (2) ]

⟹ Sum of the roots = - 6

★ Product of the roots = (1/α)(1/β)

⟹ Product of the roots = 1/αβ

⟹ Product of the roots = 1/(1/9). [ ∵ From equation (2) ]

⟹ Product of the roots = 9

Therefore,

Required quadratic equation:

⟹ x² - ( - 6)x + 9 = 0

⟹ x² + 6x + 9 = 0

∴ Option - C is the required answer.