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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
10 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
10 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
one year 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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