13. (a) On separate axes sketch the graphs of
- \(y = c ^ { 2 } - x ^ { 2 }\)
- \(y = x ^ { 2 } ( x - 3 c )\)
where \(c\) is a positive constant.
Show clearly the coordinates of the points where each graph crosses or meets the \(x\)-axis and the \(y\)-axis.
(b) Prove that the \(x\) coordinate of any point of intersection of
$$y = c ^ { 2 } - x ^ { 2 } \text { and } y = x ^ { 2 } ( x - 3 c )$$
where \(c\) is a positive constant, is given by a solution of the equation
$$x ^ { 3 } + ( 1 - 3 c ) x ^ { 2 } - c ^ { 2 } = 0$$
Given that the graphs meet when \(x = 2\)
(c) find the exact value of \(c\), writing your answer as a fully simplified surd.