8 A curve \(C\) has equation
$$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$
- Find the \(y\)-coordinates of the points on \(C\) for which \(x = 10\), giving each answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
- Sketch the curve \(C\), indicating the coordinates of any points where the curve intersects the coordinate axes.
- Write down the equation of the tangent to \(C\) at the point where \(C\) intersects the positive \(x\)-axis.
- Show that, if the line \(y = x - 4\) intersects \(C\), the \(x\)-coordinates of the points of intersection must satisfy the equation
$$16 x ^ { 2 } - 200 x + 625 = 0$$
- Solve this equation and hence state the relationship between the line \(y = x - 4\) and the curve \(C\).