If (a + bx)/(a - bx) = (b + cx)/(b - cx) = (c + dx)/(c - dx) (x ≠ 0), then show that a, b, c and d are in G.P

Solution:

It is given that (a + bx)/(a - bx) = (b + cx)/(b - cx) = (c + dx)/(c - dx) (x ≠ 0)

Therefore,

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

⇒ (a + bx)(b - cx) = (b + cx)(a - bx)

⇒ ab - acx + b²x - bcx² = ab - b²x + acx - bcx²

⇒ 2b²x = 2acx

⇒ b² = ac

⇒ b/a = c/b ....(1)

Also,

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

⇒ (b + cx)(c - dx) = (b - cx)(c + dx)

⇒ bc - bdx + c²x - cdx² = bc + bdx - c²x - cdx²

⇒ 2c²x = 2bdx

⇒ c² = bd

⇒ c/b = d/c ....(2)

From (1) and (2) , we obtain

b/a = c/b = d/c

Thus, a, b, c and d are in G.P

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NCERT Solutions Class 11 Maths Chapter 9 Exercise ME Question 13

If (a + bx)/(a - bx) = (b + cx)/(b - cx) = (c + dx)/(c - dx) (x ≠ 0), then show that a, b, c and d are in G.P

Summary:

Given that (a + bx)/(a - bx) = (b + cx)/(b - cx) = (c + dx)/(c - dx) (x ≠ 0) we found out that a, b, c and d are in a G.P by solving the equation