Question

if the root of the equation x³+ax²+bx+c=0 are in G.P. prove that a³c=b³
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Answer 1

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Step-by-step explanation:

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If roots of cubic equation are in G.P. , ax

3

+bx

2

+cx+d

Hard

Solution

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Correct option is A)

Let

Ifα,β,andγarethethreerootsthenwemusthave

ax

3

+bx

2

+cx+d=a(x−α)(x−β)(x−γ)

Comparingcoefficientsofvariouspowerofxonbothsidesleadsto

α+β+γ=−

a

b

−−−−(1)

αβ+βγ+γα=

a

c

−−−−(2)

αβγ=−

a

d

−−−(3)

Onsubstitutingβ=αrandγ=αr

2

weget

α(1+r+r

2

)=−

a

b

−−−(4)

α

2

(r+r

2

+r

3

)=

a

c

−−(5)

α

3

r

3

=−

a

d

−−−−−(6)

Now,dividing(5)by(4)toαr=−

b

c

andsubstitutingthisin(6)weget

(−

b

c

)

3

=−

a

d

⇒−

b

3

c

3

=−

a

d

∴c

3

a=b

3

d.

Hence, the option A is the correct answer.