Question

(C) 50,9
6. a, b and c are positive real numbers. If a 2 b, then :
(A) ac bc
(B) ac s bc
(C) ac
ac =​

Answer 1

Answer:

Given : a,b,c are in A.P

a

2

,b

2

,c

2

are in H.P

⟹

a

2

1

,

b

2

1

,

c

2

1

are in A.P.

then 2b=a+c ______ (1)

and

b

2

2

=

c

2

1

+

a

2

1

i.e

b

2

1

−

a

2

1

=

c

2

1

−

b

2

1

⇒

a

2

b

2

a

2

−b

2

=

c

2

b

2

b

2

−c

2

⇒

a

2

b

2

(a+b)(a−b)

=

c

2

b

2

(b−c)(b+c)

____ (1)

Now ∵a,b,c are in A.P

∴b−a=c−a

∴(1) becomes

a

2

a+b

=

c

2

b+c

⇒ac

2

+bc

2

=a

2

b+a

2

c

⇒bc

2

−a

2

b=0

⇒ac

2

−a

2

c=a

2

b−bc

2

⇒ac(c−a)+b(c−a)(c+a)=0

⇒(c−a)[ac+bc+ca]=0

⇒c=a or ab+bc+ca=0

⇒a=b=c then

b

a

+

c

b

+

a

c

=3