Question

दो धनात्मक संख्याओं a तथा b के बीच समांतर माध्य तथा गुणोत्तर माध्य का अनुपात m : n है। दर्शाइए कि a : b = `("m" + sqrt("m"^2 - "n"^2)) : ("m" - sqrt("m"^2 - "n"^2))`

Answer 1

a और b के बीच समांतर माध्य = `("a" + "b")/2`

a और b के बीच गुणोत्तर माध्य = `sqrt"ab"`

दोनों माध्यों का अनुपात m : n

अर्थात् `(("a" + "b")/2)/sqrt"ab" = "m"/"n"`

या `("a" + "b")/(2sqrt"ab") = "m"/"n"`

या `("a" + "b" + 2sqrt"ab")/("a" + "b" - 2sqrt"ab") = ("m" + "n")/("m" - "n")`

या `(sqrt"a" + sqrt"b")^2/(sqrt"a" - sqrt"b")^2 = ("m" + "n")/("m" - "n")`

वर्गमूल लेकर `(sqrt"a" + sqrt"b")/(sqrt"a" - sqrt"b") = sqrt("m" + "n")/sqrt("m" - "n")`

`((sqrt"a" + sqrt"b") + (sqrt"a" - sqrt"b"))/((sqrt"a" + sqrt"b") - (sqrt"a" - sqrt"b")) = (sqrt("m" + "n") + sqrt("m" - "n"))/(sqrt("m" + "n") - sqrt("m" - "n")`

`(2sqrt"a")/(2sqrt"b") = (sqrt("m" + "n") + sqrt("m" - "n"))/(sqrt("m" + "n") - sqrt("m" - "n")`

वर्ग करते हुए,

`"a"/"b" = (sqrt("m" + "n") + sqrt("m" - "n"))^2/(sqrt("m" - "n") - sqrt("m" - "n"))^2`

= `(("m" + "n") + ("m" - "n") + 2sqrt("m"^2 - "n"^2))/(("m" + "n") + ("m" - "n") - 2sqrt("m"^2 - "n"^2))`

= `(2"m" + 2sqrt("m"^2 - "n"^2))/(2"m" - 2sqrt("m"^2 - "n"^2))`

= `("m" + sqrt("m"^2 - "n"^2))/("m" - sqrt("m"^2 - "n"^2)`

अतः a : b = `"m" + sqrt("m"^2 - "n"^2) : "m" - sqrt("m"^2 - "n"^2)`