Consider-the-system-x-2-y-2-z-2x-2y-z-k-The-value-of-xy-zk-for-which-the-system-has-a-unique-solution-is-

Question Number 55312 by Joel578 last updated on 21/Feb/19

Consider the system

{\begin{cases} x^{2} + y^{2} = z \\ 2 x + 2 y + z = k \end{cases}

The value of x y + z k for which the system

has a unique solution is \dots

Answered by mr W last updated on 21/Feb/19

2^{2} z + 2^{2} z = {(z - k)}^{2}

z^{2} - 2 (k + 4) z + k^{2} = 0

4 {(k + 4)}^{2} - 4 k^{2} = 0

\Rightarrow k = - 2

\Rightarrow z = 2

x^{2} + y^{2} = 2

x + y = - 2

\Rightarrow x = - 1

\Rightarrow y = - 1

\Rightarrow x y + k z = 1 - 4 = - 3

Commented by Joel578 last updated on 22/Feb/19

t h a n k y o u v e r y m u c h

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