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Question
If a : b = c : d , then prove that `("a"^2 + "ab" +
"b"^2)/("a"^2 - "ab" + "b"^2) = ("c"^2 + "cd"+ "d"^2)/("c"^2 - "cd" + "d"^2)`
Solution
`"a"/"b" = "c"/"d" => "a" = "bc"/"d"`
LHS
`("a"^2 + "ab" + "b"^2)/("a"^2 - "ab" + "b"^2)`
`= (("bc"/"d")^2 + ("bc"/"d")"b" + "b"^2)/(("bc"/"d")^2 - ("bc"/"d")"b" + "b"^2)`
`= ("b"^2"c"^2 + "b"^2"cd" + "d"^2"b"^2)/("b"^2"c"^2 - "b"^2 "cd" + "d"^2"b"^2)`
`= ("b"^2("c"^2 + "cd" + "d"))/("b"^2("c"^2 - "cd" + "d"^2)) = ("c"^2 + "cd" + "d"^2)/("c"^2 - "cd" + "d"^2)` = RHS
LHS = RHS
Hence , proved.
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