5 The curve with equation \(y = \frac { x ^ { 2 } - k x + 2 k } { x + k }\) is to be investigated for different values of \(k\).
- Use your graphical calculator to obtain rough sketches of the curve in the cases \(k = - 2\), \(k = - 0.5\) and \(k = 1\).
- Show that the equation of the curve may be written as \(y = x - 2 k + \frac { 2 k ( k + 1 ) } { x + k }\).
Hence find the two values of \(k\) for which the curve is a straight line.
- When the curve is not a straight line, it is a conic.
(A) Name the type of conic.
(B) Write down the equations of the asymptotes. - Draw a sketch to show the shape of the curve when \(1 < k < 8\). This sketch should show where the curve crosses the axes and how it approaches its asymptotes. Indicate the points A and B on the curve where \(x = 1\) and \(x = k\) respectively.