9 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows the curve with equation
$$\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$$
and the straight line with equation
$$x + y = 2$$
\includegraphics[max width=\textwidth, alt={}, center]{354cbeda-d84e-433a-8834-a6f20e7e9513-05_805_1499_863_267}
- Write down the exact coordinates of the points where the curve with equation \(\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1\) intersects the coordinate axes.
- The curve is translated \(k\) units in the positive \(x\) direction, where \(k\) is a constant. Write down, in terms of \(k\), the equation of the curve after this translation.
- Show that, if the line \(x + y = 2\) intersects the translated curve, the \(x\)-coordinates of the points of intersection must satisfy the equation
$$3 x ^ { 2 } - 2 ( k + 4 ) x + \left( k ^ { 2 } + 6 \right) = 0$$
- Hence find the two values of \(k\) for which the line \(x + y = 2\) is a tangent to the translated curve. Give your answer in the form \(p \pm \sqrt { q }\), where \(p\) and \(q\) are integers.
- On Figure 3, show the translated curves corresponding to these two values of \(k\).
\end{table}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2 (for use in Question 5)}
\includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_677_1056_886_466}
\end{figure}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3 (for use in Question 9)}
\includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_798_1488_1891_274}
\end{figure}