It is given that the two curves \[y=4-x^2 \qquad \text{and} \qquad mx=k-y^2,\] where \(m>0\), touch exactly once.

  1. In each of the following four cases, sketch the two curves on a single diagram, noting the coordinates of any intersections with the axes:

    1. \(k<0\);

    2. \(0<k<16\), \(k/m<2\);

    3. \(k>16\), \(k/m>2\);

    4. \(k>16\), \(k/m<2\).

  2. Now set \(m=12\). Show that the \(x\)-coordinate of any point at which the two curves meet satisfies \[x^4-8x^2+12x+16-k=0.\] Let \(a\) be a value of \(x\) at the point where the curves touch. Show that \(a\) satisfies \[a^3-4a+3=0\] and hence find the three possible values of \(a\).

    Derive also the equation \[k=-4a^2+9a+16.\] Which of the four sketches in part (i) arise?