If $a, b, c$ are in H.P, then show that $\dfrac{a}{a-b}=\dfrac{a+c}{a-c}$.
Last updated date: 29th Mar 2023
•
Total views: 309.9k
•
Views today: 4.86k
Answer
309.9k+ views
Hint: Use the basic definition of three numbers being in H.P (harmonic progression) which is if three numbers a, b and c are in H.P then we can say that
Note: Whenever we come across such problems the key concept is to know the basic definitions of H.P, GP and AP. It will eventually help you get on the right track to reach the solution as all these have different definitions of three numbers to be in AP, GP or HP.
$\dfrac{1}{b} = \dfrac{1}{a} + \dfrac{1}{c}$
Which implies $b$ is
${\text{b = }}\dfrac{{2ac}}{{a + c}}$
Complete step by step solution:
Now it has been given that $a, b, c$ are in H.P
${\text{b = }}\dfrac{{2ac}}{{a + c}}$…………………………………………… (1)
Now we have to prove that $\dfrac{a}{{a - b}} = \dfrac{{a + c}}{{a - c}}$
Considering the LHS,
We have $\dfrac{a}{{a - b}}$
Substitute value of $b$ from equation (1) then we have
$\dfrac{a}{{a - \dfrac{{2ac}}{{a + c}}}}$
Taking LCM in the denominator part we have
$\dfrac{a}{{\dfrac{{{a^2} + ac - 2ac}}{{a + c}}}} = \dfrac{{a(a + c)}}{{{a^2} - ac}} = \dfrac{{a(a + c)}}{{a(a - c)}}$
This can be written as
$\dfrac{{a + c}}{{a - c}}$
Thus clearly LHS=RHS
Hence proved.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
