Question

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

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