4．A curve \(C\) has equation \(y = \mathrm { f } ( x )\) with \(\mathrm { f } ^ { \prime } ( x ) > 0\) ．The \(x\)－coordinate of the point \(P\) on the curve is \(a\) ．The tangent and the normal to \(C\) are drawn at \(P\) ．The tangent cuts the \(x\)－axis at the point \(A\) and the normal cuts the \(x\)－axis at the point \(B\) ．
（a）Show that the area of \(\triangle A P B\) is $$\frac { 1 } { 2 } [ \mathrm { f } ( a ) ] ^ { 2 } \left( \frac { \left[ \mathrm { f } ^ { \prime } ( a ) \right] ^ { 2 } + 1 } { \mathrm { f } ^ { \prime } ( a ) } \right)$$ （b）Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 5 x }\) and the area of \(\triangle A P B\) is \(\mathrm { e } ^ { 5 a }\) ，find and simplify the exact value of \(a\) ．