If `(logx)/(b-c)=(logy)/(c-a)=(logz)/(a-b)` , then which of the following is/are true? (a)`x y z=1` (b) `x^a y^b z^c=1` (c) `x^(b+c)y^(c+b)=1` (d) `x y z=x^a y^b z^c`

Options

A. ` xyz = 1`
B. ` x^(a)y^(b)z^(c) = 1`
C. ` x^(b+c) y^(c+a) z^(a+b) = 1`
D. ` xyz = x^(a) y^(b) z^(c)`

Answer

A, B, C, D

Explanation

Let `(log_(k)x)/(b - c) = (log_(k)y)/(c-a) = (log_(k) z)/(a-b) = p`
` rArr x = k^(p(b-c)),y=k^(p(c-a)),z = k^(p(a-b))`
` rArrxyz = k^(p(b-c))k^(p(c-a))k^(p(a-b))`
` = k^(p(b-c)+p(c-a)+p(a-b))=k^(0) = 1`
`x^(a)y^(b)z^(c) = k^(pa(b-c))k^(pb(c-a))k^(pc(a-b))=k^(0)=1`
` x^(b+c)y^(c+a)z^(a+b) = k^(p(b+c)(b-c))k^(p(c+a)(c-a))k^(p(a+b)(a-b))`
` =k^(0) = 1`