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Question

Let α1, α2, β1, β2 be the roots of ax2+bx+c=0 and px2+qx+r=0, respectively. If the system of equations α1y+α2z=0 and β1y+β2z=0 has a non - trivial solution, then


A

b2q2=acpr

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B

b2q2=prac

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C

bq=acpr

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D

b3q3=acpr

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Solution

The correct option is A

b2q2=acpr


Since, α1, α2 are the roots pf ax2+bx+c=0.
α1+α2=ba and α1α2=ca . . . (i)
Also, β1,β2 are the roots of px2+qx+r=0
β1+β2=qp and β1β2=rp . . . (ii)
Given system of equations
α1y+α2z=0
and β1y+β2z=0, has non - trivial solution.
α1α2β1β2=0 α1α2=β1β2
Applying componendo - dividendo
α1+α2α1α2=β1+β2β1β2 (α1+α2)(β1β2)=(α1α2)(β1+β2) (α1+α2)2 {(β1+β2)24β2β2}=(β1+β2)2{(α1+α2)24α1α2}
From Eqs. (i) and (ii), we get
b2a2(q2p24rp)=q2p2(b2a24ca) b2ra2p=q2cap2 b2ra=q2cp b2q2=acpr


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