Relation between Roots and Coefficients for Quadratic
If α and β ar...
Question
If α and β are the roots of a quadratic equation satisfying the conditons αβ=4 and αα−1+ββ−1=a2−7a2−4,α,β,a∈R. For what values of ′a′ will the quadratic equation have equal roots?
A
√3
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B
√5
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C
−√3
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D
−√5
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Solution
The correct option is C−√3 α(β−1)+β(α−1)(α−1)(β−1)=a2−7a2−4
⇒2αβ−(α+β)αβ−(α+β)+1=a2−7a2−4
⇒8−(α+β)5−(α+β)=a2−7a2−4 [∵αβ=4]
1+35−(α+β)=1−3a2−4⇒(α+β)=a2+1⋯(1)
For roots to be equal αβ=4⇒α2=4⇒α=±2
Using equation (1), 2α=a2+1⇒a=±√2α−1⇒a=±√3