Question

If a+b-c=5 and a2+b2+c2=29,find the value of ab-bc-ca

Answer 1

Answer:

Value of ( ab - bc - ca ) = -2

Given:

➛a + b - c = 5

➛a² + b² + c² = 29

To Find:

Value of : ab - bc - ca

Solution:

We are given,

➛a + b - c = 5

➛a² + b² + c² = 29

a + b - c = 5

Squaring on both sides,

We get,

( a + b - c )² = 5²

We know,

( a + b - c )² = a² + b² + c² + 2ab - 2bc - 2ac

➪a² + b² + c² + 2ab - 2bc - 2ac = 25

➪( a² + b² + c² ) + 2ab - 2bc - 2ac = 25

but

➪a² + b² + c² = 29 . ( given )

➪29 + 2 ( ab - bc - ac ) = 25

➪2 ( ab - bc - ac ) = 25 - 29

➪2 ( ab - bc - ac ) = -4

➪ab - bc - ac = -4 /2

➪ab - bc - ac = -4 /2 = -2

⛬ab - bc - ca = -2

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Alternative method:

We are given,

➾a + b - c = 5

but , we know

( a + b - c )² = a² + b² + c² + 2ab - 2bc - 2ac

putting given values in given equation,

we get

➪ ( 5 )² = ( a² + b² + c² ) + 2( ab - bc - ac )

➪ 25 = 29 + 2( ab - bc - ac ).

........................... a² + b² + c² = 29 ( given )

➪ 25 - 29 = 2( ab - bc - ac )

➪ -4 = 2( ab - bc - ac )

➪ -4 /2 = ab - bc - ac

⛬ -2 = ab - bc - ac

or

ab - bc - ca = -2