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if a,b,c are in H.P. then prove that a/(b+c), b/(a+c), c/(a+b) are in H.P. Please give me complete ans with steps. Thanks in advance.

if a,b,c are in H.P. then prove that a/(b+c), b/(a+c), c/(a+b) are in H.P. Please give me complete ans with steps. Thanks in advance.

Grade:12

3 Answers

SAGAR SINGH - IIT DELHI
879 Points
11 years ago

Dear student,

a, b , c  are in H.P 

  we know  that  b =  2ac /  a+ c 

 now  we have to prove that  a/ b+ c ,  b / c+ a , c / a + b are in h.p 

 i .e  we  have  to prove that  b = 2ac / a+ c ---1

 

 

 

 

 

substitute value of  a+ c  from  --1  in above equation 

 

 

 

 

 

hence proved  these  are in h.p

akshay khandelwal
18 Points
11 years ago

(b+c)/a,(a+c)/b,(a+b)/c will be in a.p.

so (b+c)/a+1,(a+c)/b+1,(a+b)/c+1 also in a.p

=(a+b+c)/a,(a+b+c)/b,(a+b+c)/c

so1/a,1/b,1/c will be in ap

so a,b,c will be in h.p                                                                                                                                                                                                                                                                                                                       

Kushagra Madhukar
askIITians Faculty 629 Points
2 years ago
Dear student,
Please find the solution to your problem below.
 
Let us assume (b+c)/a, (a+c)/b, (a+b)/c are in AP
so (b+c)/a + 1, (a+c)/b + 1, (a+b)/c + 1 also in AP
or, (a+b+c)/a, (a+b+c)/b, (a+b+c)/c are in AP
so,1/a, 1/b, 1/c are in AP
so, a, b, c are in HP
 
Hence, whenever a/(b + c), b/(a + c), c/(a + b) are in HP; a, b, c will also be in HP
Hence, vice-versa is also true thus when a, b, c are in HP; a/(b + c), b/(a + c), c/(a + b) will also be in HP
 
Thanks and regards,
Kushagra

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