Check sibling questions

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


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 9.3, 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) 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

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.