Question

Let $x,x^{lo{g}_{10}x},y^{lo{g}_{10}y}$ and $(xy{)}^{lo{g}_{10}\left(xy\right)}$ are four consecutive terms of a geometric progression (x, y > 0).

Number of ordered pair (x, y) is

• A. 2
• B. 3
• C. 4
• D. more than four

Answer 1

The correct option is B

3

If a, b, c, d are four terms in G.P. then loga, logb, logc and logd are in A.P.
$\therefore lo{g}_{10}x,lo{g}_{10}^{2}x,lo{g}_{10}^{2}y,lo{g}_{10}^2\left(xy\right)$, in A.P.
$\Rightarrow lo{g}_{10}^2\left(xy\right),-lo{g}_{10}^{2}y-lo{g}_{10}^{2}x,\dots \left(i\right)$
Also $lo{g}_{10}^{2}y-lo{g}_{10}^{2}x,=lo{g}_{10}^{2}x-lo{g}_{10}^{2}x,\dots \left(2\right)$
$\therefore form\left(1\right)$
$(lo{g}_{10}x+lo{g}_{10}y)(lo{g}_{10}x-lo{g}_{10}y)=lo{g}_{10}^{2}y-lo{g}_{10}^{2}x(lo{g}_{10}x+2lo{g}_{10}y)\left(lo{g}_{10}x\right)=lo{g}_{10}^{2}y-lo{g}_{10}^{2}x\Rightarrow lo{g}_{10}^{2}y-2lo{g}_{10}y.lo{g}_{10}x-2lo{g}_{10}^{2}x=0\dots \left(3\right)from\left(2\right)$
$lo{g}_{10}^{2}y-2lo{g}_{10}^{2}x+lo{g}_{10}x=0\dots \left(4\right)\left(3\right)-\left(4\right)\Rightarrow 2lo{g}_{10}y.lo{g}_{10}x-lo{g}_{10}x=0\Rightarrow x=1ory=\frac{1}{\sqrt{10}}$
Now, if $x=1\Rightarrow y=1$
Also, if $x=1\Rightarrow y=1$
Also, if $y=\frac{1}{\sqrt{10}}\Rightarrow 2lo{g}_{10}^{2}x=lo{g}_{10}x-\frac{1}{4}=0$
$lo{g}_{10}^{x}x=\frac{1\pm \sqrt{1+2}}{4}=\frac{1\pm \sqrt{3}}{4}\Rightarrow x=x^{\frac{1\pm \sqrt{3}}{4}}=\frac{1\pm \sqrt{3}}{4}$
$\Rightarrow x=x^{\frac{1\pm \sqrt{3}}{4}}\Rightarrow (1,1),({10}^{\frac{1+\sqrt{3}}{4}},\frac{1}{\sqrt{10}})({10}^{\frac{1-\sqrt{3}}{4}},\frac{1}{\sqrt{10}})$ are three ordered pair