Question

If \(a^x=b^y=c^z=d^u\) and \(a,b,c,d\) are in GP, then \(x,y,z,u\) are in

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

Hint:

If \(\log a,\log b,\log c,\log d\) are in AP, then their reciprocals are in HP. This is a standard result used in such exponent questions.

Answer 1

The Correct Option is C

Step 1: Let the common value be \(k\).
\[ a^x=b^y=c^z=d^u=k \] Taking logs:
\[ x\log a = y\log b = z\log c = u\log d = \log k \] So:
\[ x = \frac{\log k}{\log a},\quad y = \frac{\log k}{\log b},\quad z = \frac{\log k}{\log c},\quad u = \frac{\log k}{\log d} \] Step 2: Since \(a,b,c,d\) are in GP.
That means:
\[ b^2=ac,\quad c^2=bd \] Taking logs:
\[ 2\log b=\log a+\log c \] \[ 2\log c=\log b+\log d \] So \(\log a, \log b, \log c, \log d\) are in AP.
Step 3: Relationship of \(x,y,z,u\).
\[ x=\frac{\log k}{\log a} \] So \(x,y,z,u\) are proportional to reciprocals of \(\log a,\log b,\log c,\log d\).
Reciprocals of an AP are in HP.
Thus \(x,y,z,u\) are in HP.
Final Answer: \[ \boxed{\text{HP}} \]