If a – B = 4 and a + B = 6; Find (I) A2 + B2 (Ii) Ab – Mathematics | Shaalaa.com

(i) We know that,
( a – b )2 = a2 – 2ab + b2 
Rewrite the above identity as,
a2  + b2 = ( a – b ) + 2ab           ….(1)
Similarly, we know that,
( a + b )2 = a2 + 2ab + b2
Rewrite the above identity as,
 a2  + b2 = ( a + b )2 – 2ab                                     …..(2)
Adding the equations (1) and (2), we have
2( a2 + b2 ) = ( a – b )2 + 2ab + ( a + b )2 – 2ab
⇒ 2( a2 + b2 ) = ( a – b )2  + ( a + b )2
⇒ ( a2 + b2 ) = `1/2[( a – b )^2  + ( a + b )^2]`          ….(3)

Given that a + b = 6 ; a – b = 4
Substitute the values of ( a + b ) and (a – b)
in equation (3), we have
a2 + b2 = `1/2[ (4)^2 + (6)^2]`

= `1/2[ 16 + 36 ]`

= `52/2`
⇒ ( a2 + b2 ) = 26                                            …..(4)

From equation (4), we have
a2 + b2 = 26
Consider the identity,
( a – b )2 = a2 + b2 – 2ab                                ….(5)
Substitute the value a – b = 4 and a2 + b2 = 26
in the above equation, we have
(4)2 = 26 – 2ab
⇒ 2ab = 26 – 16
⇒  2ab = 10
⇒  ab = 5