Ex 9.3, 28 - Sum of two numbers is 6 times their geometric mean - Geometric Mean (GM)

Ex 9.3, 28 - Chapter 9 Class 11 Sequences and Series - Part 2
Ex 9.3, 28 - Chapter 9 Class 11 Sequences and Series - Part 3 Ex 9.3, 28 - Chapter 9 Class 11 Sequences and Series - Part 4 Ex 9.3, 28 - Chapter 9 Class 11 Sequences and Series - Part 5

This video is only available for Teachoo black users

This video is only available for Teachoo black users

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex9.3, 28 The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3 + 2 2) :"(3 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 Ex 8.2, 28 The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3 + 2 2) :"(3 2 " 2) Let a & b be the numbers We know that Geometric mean of two numbers a & b is i.e. GM of a & b = According to the question Sum of two numbers a and b is 6 times of their GM a + b = 6 Solving, ( + )/(2 ) = 3/1 Applying componendo & dividendo ( + +2 )/( + 2 ) = (3 + 1)/(3 1 ) (( )2+( )2+2( ))/(( )2+( )2 2( ) ) = 4/2 Using (x + y)2 = x2 + y2 + 2xy (x - y)2 = x2 + y2 - 2xy ( + )2/( )2 = 2/1 (( + )/( ))^2 = 2/1 ( + )/( ) = 2/( 1) Again applying componendo & dividendo (( + )+( ))/(( + ) ( ) ) = ( 2 + 1)/( 2 1) ( + + )/( + + ) = ( 2 + 1)/( 2 1) (2 + 0)/( + + ) = ( 2 + 1)/( 2 1) (2 )/(2 + 0) = ( 2 + 1)/( 2 1) (2 )/(2 ) = ( 2 + 1)/( 2 1) ( / ) = ( 2 + 1)/( 2 1) Squaring both sides ( ( / ))^2 = (( 2 + 1)/( 2 1))^2 / = (( 2 + 1)2)/(( 2 1)2) / = (( 2)2 + (1)2 + 2 2 1)/(( 2)2 + (1)2 2 2 1) / = (2 + 1 + 2 2)/(2 + 1 2 2) / = (3 + 2 2)/(3 2 2) Thus the ratio of a & b is 3 + 2 3: 3 2 2 Hence proved

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.