3. Find
the values of a and b for which
the roots of the equation ax^4-16x^3+ax^2+bx-7=0 are in
arithmetical progression.
Answers
Answered by
0
Let the roots be k,kr,kr
2
k+kr+kr
2
=−a=>k(1+r+r
2
)=−a ....(i)
k.kr+k.kr
2
+kr.kr
2
=b=>k
2
r(1+r+r
2
)=b ...(ii)
k.kr.kr
2
=27=>(kr)
3
=27=>kr=3 ...(iii)
From (ii): 3k(1+r+r
2
)=b ...(iv)
From (i) and (iv): b=−3a
Thus, a+b+6=0=>a=3,b=−9....(v)
From (iii): k=
r
3
...(vi)
Using (vi) and (v) in (i):
r
3
(1+r+r
2
)=−3
=>1+r+r
2
=−r
=>r
2
+2r+1=0
=>(r+1)
2
=0=>r=−1
=>k=
r
3
=−3
Hence, (a) and (b) are correct.
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