Question

If $\frac{a+bx}{a-bx}=\frac{b+cx}{b-cx}=\frac{c+dx}{c-dx},(x\ne 0)$ , then $a,b,c,d$ are in

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is B

GP

Explanation for the correct option:

Step 1. Take $\frac{a+bx}{a-bx}=\frac{b+cx}{b-cx}$

By cross multiplying, we get

$(a+bx)(b-cx)=(b+cx)(a-bx)$

$\Rightarrow$$ab+b^{2}x-acx-bc{x}^2=ab+acx-b^{2}x-bc{x}^2$

$\Rightarrow$ $2b^{2}x=2acx$

$\Rightarrow$ $b^2=ac$ …..(1)

Step 2. Take $\frac{b+cx}{b-cx}=\frac{c+dx}{c-dx}$

By cross multiplying, we get

$(b+cx)(c-dx)=(c+dx)(b-cx)$

$\Rightarrow$$bc+c^{2}x-bdx-cd{x}^2=bc+bdx-c^{2}x-cd{x}^2$

$\Rightarrow$ $2c^{2}x=2bdx$

$\Rightarrow$ $c^2=bd$ …..(2)

$\therefore$From equation(1) and (2) we can conclude that $a,b,c,d$ are in G.P

Hence, Option ‘B’ is Correct.