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If one zero of the polynomial $a 2 + 9 x 2 + 13 x + 6 a$ is reciprocal of the other, find the value of $a$ .

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1

2

3

4

answer is C.

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Detailed Solution

Given that the polynomial

a 2 + 9 x 2 + 13 x + 6 a

has one zero reciprocal of the other zero. We need to find the value of

a

.
When

α

and

β

are the zeroes of quadratic polynomial, the two equations formed with zeroes are given as:

α + β = − b a

and

α β = c a

where

b

is the coefficient of linear term,

c

is constant while

a

is the coefficient of quadratic term in quadratic polynomial.
Let us assume that one of the zeroes of the polynomial is

α .

Therefore, the other zero will be

1 α .

Now, for polynomial

a 2 + 9 x 2 + 13 x + 6 a

we have,

α + 1 α = − 13 a 2 + 9

α × 1 α = 6 a a 2 + 9 ... (i)

From equation

(i)

, we have,

1 = 6 a a 2 + 9 ⇒ a 2 + 9 = 6 a ⇒ a 2 − 6 a + 9 = 0 ... (i i)

Now factorizing equation

(i i)

we have,

a 2 − 6 a + 9 = 0 ⇒ a 2 − 3 a − 3 a + 9 = 0 ⇒ a (a − 3) − 3 (a − 3) = 0 ⇒ (a − 3) (a − 3) = 0 ∴ a = 3, 3

Hence the value of

a

3

.
Therefore, option 3 is correct.

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