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Solve! - Complex numbers and quadratic equations - BITSAT

Let \alpha and \beta be the roots of equation px^{2}+qx+r=0,p\neq 0.\; if\; p,q,r are in A.P. and \frac{1}{\alpha }+\frac{1}{\beta }=4,   then the value of \left | \alpha -\beta \right | is ?

  • Option 1)

    \frac{\sqrt{34}}{9}

  • Option 2)

    \frac{2\sqrt{13}}{9}

  • Option 3)

    \frac{\sqrt{61}}{9}

  • Option 4)

    \frac{2\sqrt{17}}{9}

 
Answers (2)
124 Views
N neha

As we have learned

Sum of Roots in Quadratic Equation -

\alpha +\beta = \frac{-b}{a}

- wherein

\alpha \: and\beta are root of quadratic equation

ax^{2}+bx+c=0

a,b,c\in C

 

 

Product of Roots in Quadratic Equation -

\alpha \beta = \frac{c}{a}

- wherein

\alpha \: and\ \beta are roots of quadratic equation:

ax^{2}+bx+c=0

a,b,c\in C

 

 @1449 

|\alpha -\beta | = \left | \frac{\sqrt{q^2}-4pr}{p} \right |

\left ( \because \left | \frac{\sqrt{D}}a{} \right | \right )

Also \frac{\alpha +\beta }{\alpha \beta }= 4

\Rightarrow \frac{-q}{r}= 4

\Rightarrow q = -4r ....(1)

= \sqrt{16(\frac{r}{p})^2-(4\frac{r}{p})}

Also p+r =2q 

\Rightarrow p+r = -8r \Rightarrow r/p = -1/9

\therefore \frac{\left | \alpha -\beta \right |}{16\times 1/81+4/9}= \sqrt{\frac{52}{81}}=\frac{2\sqrt{13}}{9}

 

 

 

 

 

 


Option 1)

\frac{\sqrt{34}}{9}

Option 2)

\frac{2\sqrt{13}}{9}

Option 3)

\frac{\sqrt{61}}{9}

Option 4)

\frac{2\sqrt{17}}{9}

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