Question

Three unequal positive numbers $a, b, c$ are such that $a, b, c$ are in G.P. while $\log \left(\frac{5 c}{2 a}\right), \log \left(\frac{7 b}{5 c}\right), \log \left(\frac{2 a}{7 b}\right)$ are in A.P. Then $a, b, c$ are the lengths of the sides of

• A. an isosceles triangle
• B. an equilateral triangle
• C. a scalene triangle
• D. a right-angled triangle

Answer 1

Answer: (C)

$\log \left(\frac{5 c}{2 a}\right)+\log \left(\frac{7 b}{5 c}\right)=\log \left(\frac{2 a}{7 b}\right) $
$\Rightarrow \log \frac{5 c}{7 b}=\log \frac{49 b^{2}}{25 c^{2}}$
$\Rightarrow 5^{3} c^{3}=7 b^{3} $
$\Rightarrow 5 c=7 b $
$\Rightarrow c=\frac{7}{5} b$
$\because b^{2}=a c=a \cdot \frac{7}{5} b$
$ \Rightarrow a=\frac{5 b}{7}$
Sides are $\frac{5 b}{7}, b, \frac{7}{5} b$