Question

If $ a, $ $ b, $ $ c $ are in $ H.P. $ , then the value of $ \dfrac{{a + c}}{{a - c}} $ is
(A) $ \dfrac{a}{{a - b}} $
(B) $ \dfrac{{a - b}}{a} $
(C) $ \dfrac{b}{a} $
(D) $ \dfrac{a}{{a + b}} $

Answer 1

Hint: Use the property that $ H.P. $ is reciprocal of $ A.P. $ . Hence, write the terms in reciprocal to find relation between $ a, $ $ b, $ $ c $ using properties of $ A.P. $ . Then substitute the values that you obtain into the expression given in the question.

Complete step-by-step answer:
We know that, if $ a, $ $ b, $ $ c $ are in $ H.P. $
Then $ \dfrac{1}{a}, $ $ \dfrac{1}{b}, $ $ \dfrac{1}{c} $ are in $ A.P. $
And we also know that,
If $ a, $ $ b, $ $ c $ are in $ A.P. $ then
 $ 2b = a + c $
Therefore, using this property of $ A.P. $ , we can write
 $ \dfrac{2}{b} = \dfrac{1}{a} + \dfrac{1}{c} $ . . . (1)
 $ \dfrac{1}{c} = \dfrac{2}{b} - \dfrac{1}{a} $
Subtract both the sides by $ \dfrac{1}{a} $ so that we could convert it to the expression similar to the question.
  $ \Rightarrow \dfrac{1}{c} - \dfrac{1}{a} = \dfrac{2}{b} - \dfrac{2}{a} $ . . . (2)
Now, we need to find the value of $ \dfrac{{a + c}}{{a - c}} $
We first need to write it in terms of the equations that we have formed. So that we could simplify the question. For that, divide the numerator and denominator of the above expression by $ ac $
Thus we get
 $ \dfrac{{a + c}}{{a - c}} = \dfrac{{\dfrac{{a + c}}{{ac}}}}{{\dfrac{{a - c}}{{ac}}}} $
 $ = \dfrac{{\dfrac{a}{{ac}} + \dfrac{c}{{ac}}}}{{\dfrac{a}{{ac}} - \dfrac{c}{{ac}}}} $
By cancelling the common terms, we get
 $ = \dfrac{{\dfrac{1}{c} + \dfrac{1}{a}}}{{\dfrac{1}{c} - \dfrac{1}{a}}} $
Now substituting the value of $ \dfrac{1}{a} + \dfrac{1}{c} $ and $ \dfrac{1}{c} - \dfrac{1}{a} $ in above expression from equation (1) and (2), we get
 $ \dfrac{{\dfrac{1}{c} + \dfrac{1}{a}}}{{\dfrac{1}{c} - \dfrac{1}{a}}} = \dfrac{{\dfrac{2}{b}}}{{\dfrac{2}{b} - \dfrac{2}{a}}} $
Cancelling $ 2 $ from the numerator and denominator, we get
 $ = \dfrac{{\dfrac{1}{b}}}{{\dfrac{1}{b} - \dfrac{1}{a}}} $
By doing cross multiplication in the denominator, we get
 $ = \dfrac{{\dfrac{1}{b}}}{{\dfrac{{a - b}}{{ab}}}} $
We know that $ \dfrac{{\dfrac{a}{b}}}{{\dfrac{x}{y}}} $ can be written as $ \dfrac{a}{b} \times \dfrac{y}{x} $
Using this property, we can simplify our expression as
 $ = \dfrac{a}{{a - b}} $
Thus,
 $ \dfrac{{a + c}}{{a - c}} = \dfrac{a}{{a - b}} $
Therefore, from the above explanation, the correct answer is option (A) $ \dfrac{a}{{a - b}} $

Note: We do not have many properties of $ H.P. $ That is why we always write the terms of $ H.P. $ into $ A.P. $ so that we could use the properties of $ A.P. $ to solve the question. Since, $ A.P. $ is reciprocal of $ H.P. $ you should know that you will form expressions in reciprocal terms. Keep that in mind and try to convert the expression in the form of reciprocals. Just like we did in this question. We converted the expression in the question in terms of reciprocals so that we could use the equations that we formed using properties of $ A.P. $