Question

if a,b,c are in A.P if the equation (b-c)x^2+(c-a)x+(a-b)=0 and 2 (c+a)x^2+(b+c)x=0 have a common root then a^2 c^2/b^4 is divisible by

Answer 1

Answer:

Given a,b,c are in AP

2b=a+c

(b−c)x

2

+(c−a)x+(a−b)=0

(b−c)x

2

−(b−c)x−(a−b)x+(a−b)=0

(b−c)x(x−1)−(a−b)(x−1)=0

(x−1)[(b−c)x−(a−b)]=0

x=

b−c

a−b

,1

x=

a−b

a−b

(∵2b=a+cb−a=c−b)

x=1

Now 2(c+a)x

2

+(b+c)x=0

x[2(c+a)x+(b+c)]=0

x=0

x=

2.2b

−(b+c)

x=−

4b

b+c

Now

4b

−(b+c)

=1

−4b=b+c

5b=−c

a+c=2b

a−5b=2b

a=7d

c

2

−a

2

=25b

2

−49b

2

=−24b

2

b

2

−c

2

=b

2

−25b

2

=−24b

2

So,

c

2

−a

2

=b

2

−c

2

hence a

2

,c

2

,b

2

are in Ap