Question:hard

What is the value of xyz?

Statement 1: \( x^{a} = y^{b} = z^{c} \) and \( ab + bc + ca = 0 \) where a, b and c are non-zero integers
Statement 2: \( a^{x} = b, \, b^{y} = c, \, c^{z} = a \) where a, b and c are non-zero integers

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Try writing all three quantities in terms of one common base or one common exponent.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is D

Solution and Explanation

Step 1: Switch to logarithms for a cleaner path.
Taking logs turns products of powers into sums, which is often easier to track.

Step 2: Apply logs to statement 1.
From \( x^{a} = y^{b} = z^{c} \), take natural logs: \( a \ln x = b \ln y = c \ln z = L \) for some constant L.
Then \( \ln x = L/a \), \( \ln y = L/b \), \( \ln z = L/c \).
Adding gives \( \ln(xyz) = L \left( \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \right) = L \cdot \dfrac{ab+bc+ca}{abc} \).
Because \( ab + bc + ca = 0 \), this sum is 0, so \( \ln(xyz) = 0 \) and \( xyz = 1 \). Statement 1 alone works.

Step 3: Apply logs to statement 2.
Take logs of each equation: \( x \ln a = \ln b \), \( y \ln b = \ln c \), \( z \ln c = \ln a \).
Multiply all three left sides and all three right sides: \( xyz \cdot (\ln a)(\ln b)(\ln c) = (\ln b)(\ln c)(\ln a) \).
The factor \( (\ln a)(\ln b)(\ln c) \) is common and nonzero, so it cancels, leaving \( xyz = 1 \).
Statement 2 alone works too, confirmed by a completely different route.

Final Answer:
Both routes agree that either statement alone fixes xyz = 1. \[ \boxed{(d)} \]
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