Question

If $a,b,c$ are in G.P. and $log\left(\frac{5c}{a}\right),log\left(\frac{3b}{5c}\right),log\left(\frac{a}{3b}\right)$ are in A.P., then the number $a,b,c$ form an

• A. Equilateral triangle
• B. Isosceles triangle
• C. Right angle triangle
• D. None of these

Answer 1

The correct option is D None of these

Given a,b,c $ϵ$ G.P.$\therefore b^2=ac...(\ast )$

and $log\left(\frac{5c}{a}\right),log\left(\frac{3b}{5c}\right),log\left(\frac{a}{3b}\right)$ are in A.P

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

$\Rightarrow {\left(\frac{3b}{5c}\right)}^2=\frac{5ac}{3ba}$

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

or $(3b{)}^3=(5c{)}^3$

or $\frac{b}{c}=\frac{5}{3}orc=\frac{3b}{5}$

or $\frac{b^2}{c^2}=\frac{25}{9}or\frac{ac}{c^2}=\frac{25}{9}$

$\Rightarrow \frac{a}{c}=\frac{25}{9}ora=\frac{25}{9}$

$=\frac{25}{9}\times \frac{3b}{5}=\frac{5}{3}b$

$\therefore$ number are

$\frac{5}{3}b,b,\frac{3}{5}bor25,15,9$

Which do not form any triangle as sum of two number is less than the third.