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प्रश्न
If a - b = 4 and a + b = 6; find
(i) a2 + b2
(ii) ab
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उत्तर
(i) We know that,
( a - b )2 = a2 - 2ab + b2
Rewrite the above identity as,
a2 + b2 = ( a - b )2 + 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
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