Step-by-step explanation:
To find the value of \( ab + bc - ca \), we can first solve the given equations:
Given:
1. \( a - b + c = 6 \)
2. \( a^2 + b^2 + c^2 = 38 \)
We can square the first equation to eliminate the square roots:
\((a - b + c)^2 = 6^2 \)
\(a^2 + b^2 + c^2 - 2ab + 2ac - 2bc = 36 \)
Now, we can substitute the value of \(a^2 + b^2 + c^2\) from the second equation:
\(38 - 2ab + 2ac - 2bc = 36 \)
\(2ab - 2ac + 2bc = 2 \)
\(ab - ac + bc = 1 \)
Therefore, \(ab + bc - ca = 1\).
So, the answer is (b) 1.
Answer:
[tex]\boxed{\bf\:(b) \: \: 1 \: } \\ [/tex]
Step-by-step explanation:
Given that,
[tex]\sf\: a - b + c = 6 \\ [/tex]
[tex]\sf\: {a}^{2} + {b}^{2} + {c}^{2} =38 \\ [/tex]
Now, Consider
[tex]\sf\: a - b + c = 6 \\ [/tex]
On squaring both sides, we get
[tex]\sf\: (a - b + c)^{2} = 36 \\ [/tex]
We know,
[tex]\boxed{\sf\: {(x + y + z)}^{2} = {x}^{2} + {y}^{2} + {z}^{2} + 2xy + 2yz + 2zx \: } \\ [/tex]
So, using this algebraic identity, we get
[tex]\sf\: (a)^{2} + {( - b)}^{2} + {(c)}^{2} + 2(a)( - b) + 2( - b)(c) + 2ca = 36 \\ [/tex]
[tex]\sf\: a^{2} + {b}^{2} + {c}^{2} - 2ab - 2bc + 2ca = 36 \\ [/tex]
On substituting the value from given, we get
[tex]\sf\: 38 - 2ab - 2bc + 2ca = 36 \\ [/tex]
[tex]\sf\: - 2ab - 2bc + 2ca = 36 - 38 \\ [/tex]
[tex]\sf\: - 2ab - 2bc + 2ca = - 2 \\ [/tex]
[tex]\sf\: - 2(ab + bc - ca) = - 2 \\ [/tex]
[tex]\implies\sf\:ab + bc - ca= 1 \\ [/tex]
Hence,
[tex]\implies\sf\:\boxed{\bf\:ab + bc - ca= 1 \: } \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
[tex]\bf\: Additional\: Information:[/tex]
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]
