Question

If $a,b,c$ are in $H.P.$, then which one of the following is true ?

• A. $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{b}$
• B. $\frac{ac}{a+c}=b$
• C. $\frac{b+a}{b-a}+\frac{b+c}{b-c}=1$
• D. $Noneofthese$

Answer 1

Answer: (D)

$Noneofthese$

a,b,c are in HP

$\frac{2}{b}=\frac{1}{a}+\frac{1}{c}$

$\frac{2}{b}=\frac{a+c}{ac}$

$\frac{b}{2}=\frac{ac}{a+c}\Rightarrow \frac{2ac}{a+c}$

1) $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{b}$

$\frac{b-c+b-a}{(b-c)(b-a)}=\frac{1}{b}$

$(2b-a-c)b=b^2-ab-cb+ac$

$2b^2-ab-bc=b^2-ab-b^{2}c+ac$

$b^2=ac$

False

2) $b=\frac{ac}{a+c}$ False

3) $\frac{b+a}{b-a}+\frac{b+c}{b-c}=1$

$(b+a)(b-c)+(b+c)(b-a)=(b-a)(b-c)$

$b^2-bc+ab-ac+b^2-ab+bc-ac=b^2-ab-bc+ac$

$b^2-2ac=-ab-bc+ac$

$b^2=\frac{-ab-bc+ac}{2ac}$

False

$\therefore$None of the above