Question

If $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$ , then which of the following is/are true?

• A. $\mathrm{xyz}=1$
• B. $x^a{y}^b{z}^c=1$
• C. $x^a{y}^b{z}^c=\mathrm{xyz}$
• D. $x^{b+c}{y}^{c+a}{z}^{a+b}=1$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $\mathrm{xyz}=1$
B $x^a{y}^b{z}^c=1$
C $x^a{y}^b{z}^c=\mathrm{xyz}$
D $x^{b+c}{y}^{c+a}{z}^{a+b}=1$

$\mathrm{Let}\frac{{\log }_{t}x}{b-c}=\frac{{\log }_{t}y}{c-a}=\frac{{\log }_{t}z}{a-b}=p\Rightarrow {\log }_{t}x=p\left(b-c\right),{\log }_{t}y=p\left(c-a\right),{\log }_{t}z=p\left(a-b\right)\Rightarrow x=t^{p(b-c)},y=t^{p(c-a)},z=t^{p(a-b)}\therefore \mathrm{xyz}=t^{p(b-c)}\cdot t^{p(c-a)}\cdot t^{p(a-b)}\Rightarrow \mathrm{xyz}=t^{p\left(b-c+c-a+a-b\right)}=t^0=1x^a{y}^b{z}^c=t^{\mathrm{ap}(b-c)}\cdot t^{\mathrm{bp}(c-a)}\cdot t^{\mathrm{cp}(a-b)}\Rightarrow x^a{y}^b{z}^c=t^{p\left(\mathrm{ab}-\mathrm{ac}+\mathrm{bc}-\mathrm{ab}+\mathrm{ac}-\mathrm{bc}\right)}=t^0=1\therefore x^a{y}^b{z}^c=\mathrm{xyz}{x}^{b+c}{y}^{c+a}{z}^{a+b}=t^{p(b-c)(b+c)}\cdot t^{p(c-a)(c+a)}\cdot t^{p(a-b)(a+b)}\Rightarrow x^{b+c}{y}^{c+a}{z}^{a+b}=t^{p\left(b^2-c^2+c^2-a^2+a^2-b^2\right)}=t^0=1=\mathrm{xyz}\therefore x^{b+c}{y}^{c+a}{z}^{a+b}=1$