Question

(ii) If (a+b)/(b+c) = (c+d)/(d+a), then prove that c=a or a+b+c+d=0.​

Answer 1

Question : -

If (a+b)/(b+c) = (c+d)/(d+a), then prove that c = a or a + b + c + d = 0

ANSWER

Given : -

(a+b)/(b+c) = (c+d)/(d+a)

Required to prove : -

• c = a

Proof : -

Let;

(a+b)/(b+c) = (c+d)/(d+a) = k

Here,

'k' is an constant of the ratio

So,

(a+b)/(b+c) = k

a+b = k(b+c)

a+b = bk + ck

a - c = bk - b

a - c = b(k - 1)

(a - c)/(b) = (k - 1) ....(1)

Similarly,

(c+d)/(d+a) = k

c+d = k(d+a)

c+d = dk + ak

c - a = dk - d

c - a = d(k-1)

(c - a)/(d) = (k-1) ...(2)

Compare 1 & 2

(a - c)/(b) = (c - a)/(d)

(a - c)d = (c - a)b

ad - cd = cb - ab

ad + ab = cb + cd

a(b + d) = c(b + d)

=> a = c