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Question: If a, b, c are in H.P. , then the value of\(\frac{b + a}{b - a} + \frac{b + c}{b - c}\)is...

If a, b, c are in H.P. , then the value ofb+ab−a+b+cb−c\frac{b + a}{b - a} + \frac{b + c}{b - c}is

A

0

B

1

C

2

D

3

Answer

2

Explanation

Solution

a, b, c are in H.P. ⇒ b = 2aca+c\frac{2ac}{a + c}⇒ ba=2ca+c\frac{b}{a} = \frac{2c}{a + c}

⇒ b+ab−a=3c+ac−a\frac{b + a}{b - a} = \frac{3c + a}{c - a}.……(1)

Again a, b, c are in H.P.

⇒ b = 2aca+c\frac{2ac}{a + c} ⇒ bc=2aa+c\frac{b}{c} = \frac{2a}{a + c}

⇒ b+cb−c=3a+ca−c\frac{b + c}{b - c} = \frac{3a + c}{a - c}.……(2)

From (1) and (2),

b+ab−a+b+cb−c=3c+ac−a+3a+ca−c\frac{b + a}{b - a} + \frac{b + c}{b - c} = \frac{3c + a}{c - a} + \frac{3a + c}{a - c}=2.

Hence (3) is the correct answer.

Alternative solution:

We have b+ab−a+b+cb−c=2a+2b2a−2b+2c+2b2c−2b\frac{b + a}{b - a} + \frac{b + c}{b - c} = \frac{\frac{2}{a} + \frac{2}{b}}{\frac{2}{a} - \frac{2}{b}} + \frac{\frac{2}{c} + \frac{2}{b}}{\frac{2}{c} - \frac{2}{b}}

= 3a+1c1a−1c+3c+1a1c−1a=2a−2c1a−1c=2\frac{\frac{3}{a} + \frac{1}{c}}{\frac{1}{a} - \frac{1}{c}} + \frac{\frac{3}{c} + \frac{1}{a}}{\frac{1}{c} - \frac{1}{a}} = \frac{\frac{2}{a} - \frac{2}{c}}{\frac{1}{a} - \frac{1}{c}} = 2