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 \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) 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

Ans: (C)
Hint \(: \log \frac{5 c}{2 a}+\log \frac{2 a}{7 b}=2 \log \frac{7 b}{5 c} \Rightarrow \log \frac{5 c}{7 b}=\log \frac{49 b^{2}}{25 c^{2}}\)
\(\Rightarrow 5^{3} c^{3}=7 b^{3} \Rightarrow 5 c=7 b \Rightarrow c=\frac{7}{5} b\)
\(\because b^{2}=a c=a \cdot \frac{7}{5} b \Rightarrow a=\frac{5 b}{7}\)
Sides are \(\frac{5 b}{7}, b, \frac{7}{5} b\)

Answer 2

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