Question : If $x+y+z=13$ and $x^2+y^2+z^2=69$, then $xy+z(x+y)$ is equal to:
Option 1: 70
Option 2: 40
Option 3: 50
Option 4: 60
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Correct Answer: 50
Solution :
Given: $x+y+z=13$ and $x^2+y^2+z^2=69$.
To find: $xy+z(x+y)$ which is equal to $(xy+zx+zy)$
Squaring both sides,
$(x+y+z)^{2}=13^2$
$x^{2}+y^{2}+z^{2}+2(xy+yz+zx)=169$
Putting the values, we get,
$69+2(xy+yz+zx)=169$
⇒ $2(xy+yz+zx)=169-69$
⇒ $(xy+yz+zx) = \frac{100}{2} = 50$
Hence, the correct answer is 50.
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