If 2^x=3^y=12^z,then prove that- xy=z(x+2y)

Asked by sajal.dada09 | 19th May, 2019, 08:13: PM

Expert Answer:

begin mathsize 16px style Given colon space 2 to the power of straight x space equals space 3 to the power of straight y space equals space 12 to the power of straight z space
Let space space 2 to the power of straight x space equals space 3 to the power of straight y space equals space 12 to the power of straight z space space equals space straight k
rightwards double arrow space 2 space equals space straight k to the power of 1 over straight x end exponent space comma space 3 space equals space straight k to the power of 1 over straight y end exponent space and space 12 equals space straight k to the power of 1 over straight z end exponent.... space left parenthesis straight i right parenthesis
We space know space that comma

12 space equals space 2 squared space cross times space 3
rightwards double arrow space space straight k to the power of 1 over straight z end exponent space equals straight k to the power of 2 over straight x end exponent space cross times space straight k to the power of 1 over straight y end exponent
rightwards double arrow space straight k to the power of 1 over straight z end exponent space equals straight k to the power of fraction numerator 2 straight y space plus space straight x over denominator xy end fraction end exponent space
rightwards double arrow 1 over straight z space equals fraction numerator straight x space plus space 2 straight y over denominator xy end fraction space
rightwards double arrow xy space equals straight z open parentheses straight x space plus space 2 straight y close parentheses space
Hence space proved. end style

Answered by Yasmeen Khan | 20th May, 2019, 09:54: AM