Question

If a, b, c are in continued proportion the expression $\frac{a^2+ab+b^2}{b^2+bc+c^2}$ can be simplified to

• A. $\frac{a+c}{a+b}$
• B. $\frac{a}{c}$
• C. $\frac{c}{a}$
• D. $\frac{a-c}{b}$

Answer 1

Answer: (B)

$\frac{a}{c}$

$\frac{a^2+ab+b^2}{b^2+bc+c^2}$
a, b, c are in continued proportion.
$\therefore a=a,b=ak,c=a{k}^2$
$\therefore \frac{a^2+ab+b^2}{b^2+bc+c^2}$
$=\frac{a^2+a\left(ak\right)+(ak{)}^2}{(ak{)}^2+(ak{)}^2\left(a{k}^2\right)+(a{k}^2{)}^2}$
$=\frac{a^2+a^{2}k+a^2{k}^2}{a^2{k}^2+a^2{k}^3+a^2{k}^4}$
$=\frac{a^2(1+k+k^2}{a^2{k}^2(1+k+k^2}$
$=\frac{1}{k^2}$
$=\frac{a}{a{k}^2}$
$=\frac{a}{c}$