find the condition that the equation x^3+3px^2+3qx+r=0 maybe in G.P
Answers
Answer:
Given that the roots of x
3
+3px
2
+3qx+r=0 are in H.P. (1)
Let x=
x
1
⇒(
x
1
)
3
+3p(
x
1
)
2
+3q(
x
1
)+r=0
Multiplying by x
3
throughout; we get
x
3
(
x
1
)
3
+x
3
3p(
x
1
)
2
+x
3
3q(
x
1
)+rx
3
=0
⇒rx
3
+3qx
2
+3px+1=0 (2)
The roots of the equation (2) being reciprocal of the roots of the equation (1) must be in A.P.
Let the roots of eq. (2) be α−β,α,α+β
∴ from equation (2); we get
sum of the roots=α−β+α+α+β=−
r
3q
⇒3α=−
r
3q
⇒α=−
r
q
Since α is a root of (2)
∴r(
r
−q
)
3
+3q(
r
−q
)
2
+3p(
r
−q
)+1=0
⇒
r
2
−q
3
+
r
2
3q
3
−
r
3pq
+1=0
⇒2q
3
−3pqr+r
2
=0
∴ The required condition is 2q
3
−3pqr+r
2
=0
Answer:
Given that the roots of x
3
+3px
2
+3qx+r=0 are in H.P. (1)
Let x=
x
1
⇒(
x
1
)
3
+3p(
x
1
)
2
+3q(
x
1
)+r=0
Multiplying by x
3
throughout; we get
x
3
(
x
1
)
3
+x
3
3p(
x
1
)
2
+x
3
3q(
x
1
)+rx
3
=0
⇒rx
3
+3qx
2
+3px+1=0 (2)
The roots of the equation (2) being reciprocal of the roots of the equation (1) must be in A.P.
Let the roots of eq. (2) be α−β,α,α+β
∴ from equation (2); we get
sum of the roots=α−β+α+α+β=−
r
3q
⇒3α=−
r
3q
⇒α=−
r
q
Since α is a root of (2)
∴r(
r
−q
)
3
+3q(
r
−q
)
2
+3p(
r
−q
)+1=0
⇒
r
2
−q
3
+
r
2
3q
3
−
r
3pq
+1=0
⇒2q
3
−3pqr+r
2
=0
∴ The required condition is 2q
3
−3pqr+r
2
=0
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