If a^2+b^2+4c^2=ab+2bc+2ac, prove that a =b=2c.

Asked by aryadutta2002 | 17th Jun, 2017, 07:09: PM

Expert Answer:

begin mathsize 16px style straight a squared plus straight b squared plus 4 straight c squared equals ab plus 2 bc plus 2 ac
rightwards double arrow 2 straight a squared plus 2 straight b squared plus 8 straight c squared equals 2 ab plus 4 bc plus 4 ac space space space space space space.... left parenthesis Multiplying space throughout space by space 2 right parenthesis
rightwards double arrow straight a squared plus straight a squared plus straight b squared plus straight b squared plus 4 straight c squared plus 4 straight c squared equals 2 ab plus 4 bc plus 4 ca
rightwards double arrow straight a squared plus straight a squared plus straight b squared plus straight b squared plus 4 straight c squared plus 4 straight c squared minus 2 ab minus 4 bc minus 4 ca equals 0
rightwards double arrow left parenthesis straight a squared minus 2 ab plus straight b squared right parenthesis plus left parenthesis straight b squared minus 4 bc plus 4 straight c squared right parenthesis plus left parenthesis straight a squared minus 4 ca plus 4 straight c squared right parenthesis equals 0
rightwards double arrow left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus 4 straight c right parenthesis squared plus left parenthesis straight a minus 4 straight c right parenthesis squared equals 0
If space sum space of space the space squares space is space equal space to space zero comma space then space individual space terms space are space also space equal space to space zero.
left parenthesis straight a minus straight b right parenthesis squared equals 0 comma space space space left parenthesis straight b minus 4 straight c right parenthesis squared equals 0 comma space space left parenthesis straight a minus 4 straight c right parenthesis squared equals 0
rightwards double arrow straight a minus straight b equals 0 comma space straight b minus 4 straight c equals 0 comma space straight a minus 4 straight c equals 0
rightwards double arrow straight a equals straight b comma space straight b equals 4 straight c comma space straight a equals 4 straight c
rightwards double arrow straight a equals straight b equals 4 straight c
end style

Answered by Rashmi Khot | 18th Jun, 2017, 12:01: AM

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