Question

If log (a+c), log (a+b), log (b+c) are in A.P. and a, c, b are in H.P, then the value of a+b is (given a, b, c >0)

• A. 2c
• B. 3c
• C. 4c
• D. 6c

Answer 1

The correct option is A 2c

log(a+c)+log(b+c)=2log(a+b)
$(a+c)(b+c)=(a+b{)}^2$
$\Rightarrow ab+c(a+b)+c^2=(a+b{)}^2(1)$
also, $c=\frac{2ab}{a+b}\Rightarrow 2ab=c(a+b)$
$\Rightarrow 2ab+2c(a+b)+2c^2=2(a+b{)}^2......(2)$
From (1) and (2),
$c(a+b)+2c(a+b)+2c^2=2(a+b{)}^2$
$2(a+b{)}^2-3c(a+b)-2c^2=0$
$\therefore a+b=\frac{3c\pm \sqrt{9c^2+16c^2}}{4}=\frac{3c\pm 5c}{4}=2cor-\frac{c}{2}$
$\therefore a+b=2c$ ($\because$ a, b, c > 0)