if a : b = c : d then prove that   a^2 c + a c^2 : b^2 d + b d^2 = (a+c)^3 : (b+d)^3

Asked by Ramesh | 9th Feb, 2019, 10:04: PM

Expert Answer:

begin mathsize 16px style straight a over straight b equals straight c over straight d equals straight k space left parenthesis say right parenthesis
straight a equals bk comma space straight c equals dk.... left parenthesis straight i right parenthesis
LHS
equals fraction numerator straight a squared straight c plus ac squared over denominator straight b squared straight d plus bd squared end fraction
equals fraction numerator ac open parentheses straight a plus straight c close parentheses over denominator bd open parentheses straight b plus straight d close parentheses end fraction
equals fraction numerator begin display style bk cross times dk open parentheses bk plus dk close parentheses end style over denominator begin display style bd open parentheses straight b plus straight d close parentheses end style end fraction space space space space from space left parenthesis straight i right parenthesis
equals fraction numerator straight k squared cross times straight k open parentheses straight b plus straight d close parentheses over denominator open parentheses straight b plus straight d close parentheses end fraction
equals straight k cubed space space.... left parenthesis ii right parenthesis
RHS
equals fraction numerator begin display style open parentheses straight a plus straight c close parentheses cubed end style over denominator begin display style open parentheses straight b plus straight d close parentheses cubed end style end fraction
equals open parentheses fraction numerator straight a plus straight c over denominator straight b plus straight d end fraction close parentheses cubed
equals open parentheses fraction numerator begin display style bk plus dk end style over denominator begin display style straight b plus straight d end style end fraction close parentheses cubed
equals open parentheses fraction numerator begin display style straight k open parentheses straight b plus straight d close parentheses end style over denominator begin display style straight b plus straight d end style end fraction close parentheses cubed
equals straight k cubed space space..... left parenthesis iii right parenthesis
Form space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis
fraction numerator begin display style straight a squared straight c plus ac squared end style over denominator begin display style straight b squared straight d plus bd squared end style end fraction equals fraction numerator begin display style open parentheses straight a plus straight c close parentheses cubed end style over denominator begin display style open parentheses straight b plus straight d close parentheses cubed end style end fraction
end style

Answered by Sneha shidid | 11th Feb, 2019, 09:44: AM

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