Parameter conditions for real pairs

8 written marks

Consider \(x+y=k\) and \(xy=k+2\), where \(k\) is real.

1. Show that the possible x-values satisfy \(x^2-kx+k+2=0\).

   [2]

2. Find all \(k\) for which at least one real ordered pair \((x,y)\) satisfies the system.

   [3]

3. Find all \(k\) for which there is exactly one distinct real ordered pair, and give that pair for each \(k\).

   [3]

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