1 A family of curves is given by the equation
$$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$
where the parameter \(a\) is a real number.
- On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
- \(a = - 0.1\)
- \(a = 0.5\)
(ii) State one feature of the curve for the cases \(a = - 0.5\) and \(a = - 0.1\) that is not a feature of the curve in the case \(a = 0.5\).
(iii) By using a slider for \(a\), or otherwise, write down the non-zero value of \(a\) for which the points on the curve (\textit{) all lie on a straight line.
(iv) Write down the equation of the vertical asymptote of the curve (}).
The equation of the curve (*) can be written in the form \(y = x + A + \frac { a ^ { 2 } - a } { x - 1 }\), where \(A\) is a constant.
(v) Show that \(A = 0\).
(vi) Hence, or otherwise, find the value of
$$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$
(vii) Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).- In this part of the question the value of the parameter \(a\) satisfies \(0 < a < 1\). For values of \(a\) in this range the curve intersects the \(x\)-axis at points X and Y . The point Z has coordinates \(( 0 , - 1 )\). These three points form a triangle XYZ.
- Determine, in terms of \(a\), the area of the triangle XYZ.
- Find the maximum area of the triangle XYZ.