Misc 19 - Ratio of AM and GM of a, b is m : n. Show that - AM and GM (Arithmetic Mean And Geometric mean)

  1. Class 11
  2. Important Question for exams Class 11
Ask Download

Transcript

Misc 19 The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that a : b = (m + ( ^2 ^2 )) : (m ( ^2 ^2 ) ) Introduction Componendo dividendo If / = / Applying componendo dividendo ( + )/( ) = ( + )/( ) Eg: Taking 1/2 = 4/8 (1+ 2)/(1 2) = (4 + 8)/(4 8) 3/( 1) = 12/( 4) -3 = -3 Misc 19 The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that a : b = (m + ( ^2 ^2 )) : (m ( ^2 ^2 ) ) Here, the two numbers be a and b. Arithmetic Mean =AM= (a+b)/2 & Geometric Mean=GM= ab According to the question, AM/( GM" " ) = / ( + )/(2 " " ) = / Applying componendo dividendo ( + +2 )/( + 2 ) = ( + )/( ) (( )2+( )2+2( ))/(( )2+( )2 2( ) ) =( + )/( ) Using (x + y)2 = x2 + y2 + 2xy (x - y)2 = x2 + y2 - 2xy ( + )2/( )2 = ( + )/( ) (( + )/( ))^2 = ( + )/( ) ( + )/( ) = (( + )/( )) ( + )/( ) = ( + )/( ( ) ) Applying componendo dividendo (( + ) + ( ))/(( + ) ( ) ) = ( ( + ) + ( ))/( ( + ) ( )) (2 )/(2 ) = ( ( + ) + ( ))/( ( + ) ( )) / = ( ( + ) + ( ))/( ( + ) ( )) Squaring both sides ( / )^2 = (( ( + ) + ( ))/( ( + ) ( )))^2 ( )^2/( )^2 = ( ( + ) + ( ))^2/( ( + ) ( ))^2 Using (x + y)2 = x2 + y2 + 2xy (x - y)2 = x2 + y2 - 2xy / = (( ( + ) )^2+( ( ) )^2+ 2( ( + ))( ( )))/(( ( + ) )^2+( ( ) )^2 2( ( + ))( ( )) ) / = ( + + + 2 (( + )( ) ))/( + + 2 (( + )( ) )) / = ( + + + 2 (( ^2 ^2 ) ))/( + + 2 (( ^2 ^2 ) )) / = (2 + 2 (( ^2 ^2 ) ))/(2 2 (( ^2 ^2 ) )) / = 2( + (( ^2 ^2 ) ))/2( (( ^2 ^2 ) )) / = ( + (( ^2 ^2 ) ))/( (( ^2 ^2 ) )) Thus, a : b = (m + ( ^2 ^2 )) : (m ( ^2 ^2 ) ) Hence proved

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.