Question

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

Since given nos. are in $A.P$.
$\therefore 2log \frac{3b}{5c}= log \frac{5c}{a} +log \frac{a}{3b}$
$ \Rightarrow \left(\frac{3b}{5c}\right)^{2} = \frac{5c}{a}\cdot\frac{a}{3b}$
$ \Rightarrow 3b=5c$
Also $b^{2}= ac $
$\therefore \frac{25c^{2}}{9} = ac $
$ \Rightarrow 9a=25c $
$ \therefore \frac{9a}{5} = 5c = 3b $
$ \Rightarrow \frac{a}{5}= \frac{b}{3} = \frac{c}{{9}/{5}} $
$ \Rightarrow b+c< a$
$ \therefore a,b,c$ cannot be sides of a triangle