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  • Tell me? - Complex numbers and quadratic equations - JEE Main-7

If ax^{2}+bx+c= 0 and bx^{2}+cx+a= 0 have a common root and a,b,c are non-zero real numbers, then \frac{a^{3}+b^{3}+c^{3}}{abc}  equals 

  • Option 1)

    3

  • Option 2)

    2

  • Option 3)

    1

  • Option 4)

    -1

 

Answers (1)

best_answer

ax^{2}+bx+c=0\; \;and\; \; bx^{2}+cx+a=0   have common roots , so

\\*(a^{2}-bc)^{2}=(ac-b^{2})(ab-c^{2})\\*\Rightarrow a^{4}+b^{2}c^{2}-2a^{2}bc=a^{2}bc-ac^{3}-ab^{3}+b^{2}c^{2}\\*\Rightarrow a^{4}+ab^{3}+ac^{3} =3a^{2}bc\\*\Rightarrow a^{3}+b^{3}+c^{3}=3abc

 

Condition for one common root -

\left ( {a}' c-a{c}'\right )^{2}= \left ( b{c}' -{b}'c\right )\left ( a{b}' -{a}'b\right )

- wherein

ax^{2}+bx+c=0 &

a'x^{2}+b'x+c'=0

are the 2 equations

 

 


Option 1)

3

This is correct

Option 2)

2

This is incorrect

Option 3)

1

This is incorrect

Option 4)

-1

This is incorrect

Posted by

divya.saini

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