9 A hyperbola \(H\) has equation
$$x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$$
- Find the equations of the two asymptotes of \(H\), giving each answer in the form \(y = m x\).
- Draw a sketch of the two asymptotes of \(H\), using roughly equal scales on the two coordinate axes. Using the same axes, sketch the hyperbola \(H\).
- Show that, if the line \(y = x + c\) intersects \(H\), the \(x\)-coordinates of the points of intersection must satisfy the equation
$$x ^ { 2 } - 2 c x - \left( c ^ { 2 } + 2 \right) = 0$$
- Hence show that the line \(y = x + c\) intersects \(H\) in two distinct points, whatever the value of \(c\).
- Find, in terms of \(c\), the \(y\)-coordinates of these two points.