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If ax2 + bx + c = 0 and bx2 + cx + a = 0 have a common root and abc ≠ 0, then $\frac{a^{3} + b^{3} + c^{3}}{abc}$ =

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3

1

answer is B.

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Detailed Solution

Given, ax2 + bx + c = 0 and bx2 + cx + a = 0
Find,

\frac{a^{3} + b^{3} + c^{3}}{abc}

Let,
ax2 + bx + c = 0………(i)
bx2 + cx + a = 0…..….(ii)
Let α and β be the roots common for the above equation.
The sum of the roots is given by −

\frac{coefficient of x}{coefficient of x^{2}}

and the product of roots is given by −

\frac{constant}{coefficient of x^{2}}

Therefore,
For eq. (i), α + β =

\frac{- b}{a}

and αβ =

\frac{c}{a}

For eq. (ii), α + β =

\frac{- c}{b}

and αβ =

\frac{a}{b}

As both the roots are common, the sum and the product of the root will be equal for both the equations.
Therefore, it can be interpreted as follows:

\frac{- b}{a}

\frac{- c}{b}

⇒ b2 = ac……. (iii)
⇒ c =

\frac{b^{2}}{a}

…… (iv)
⇒ a =

\frac{b^{2}}{c}

…… (v)
Again,

\frac{c}{a}

\frac{a}{b}

⇒

a^{2}

= bc…… (vi)
Now, use equation (iv) in equation (vi).

a^{2}

= bc

a^{2}

= b

(\frac{b^{2}}{a})

[∴ c = \frac{b^{2}}{a}]

a^{2}

\frac{b^{3}}{a}

a^{2} (a)

b^{3}

a^{3}

b^{3}

Now, use equation (v) in equation (vi).

a^{2}

= bc

{(\frac{b^{2}}{c})}^{2}

= b

c

[∴ a = \frac{b^{2}}{c}]

\frac{b^{4}}{c^{2}}

= bc

b^{3}

c^{3}

So, it can be observed that,
a3 = b3 = c3 ⇒ a = b = c.
Now, use these values to find the value of

\frac{a^{3} + b^{3} + c^{3}}{abc}

\frac{a^{3} + a^{3} + a^{3}}{a . a . a}

\frac{{3 a}^{3}}{a^{3}}

= 3
Hence, option (2) is the correct option.

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