Question

If $\log \left(\frac{5c}{a}\right)$, $\log \left(\frac{3b}{5c}\right)$ and $\log \left(\frac{a}{3b}\right)$ are in AP, where $a,b,c$ are in GP, then $a,b,c$ are the lengths of sides of

• A. an isosceles triangle
• B. an equilateral triangle
• C. a scalene triangle
• D. none of these

Answer 1

Answer: (D)

none of these

If the given terms are in A.P then

$2log\left(\frac{3b}{5c}\right)=log\left(\frac{5c}{a}\right)+log\left(\frac{a}{3b}\right)\Rightarrow {\left(\frac{3b}{5c}\right)}^2=\frac{5c}{3b}\Rightarrow {\left(3b\right)}^3={\left(5c\right)}^3\Rightarrow 3b=5c\Rightarrow b=\frac{5c}{3}$

also since $a,b,c$ are in G.P

$b^2=ac$

$\Rightarrow a=\frac{b^2}{c}=\frac{25c}{9}$

so we have $a=\frac{25c}{9}$

$b=\frac{5c}{3}andc=c$

we can clearly see that $a>b+c$ which is not the property of a triangle .

Hence $a,b,c$ are not the sides of a triangle.

so the correct option is D