7. The ellipse \(E\) has foci at the points \(( \pm 3,0 )\) and has directrices with equations \(x = \pm \frac { 25 } { 3 }\)
- Find a cartesian equation for the ellipse \(E\).
The straight line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
- Show that the \(x\) coordinates of any points of intersection of \(l\) and \(E\) satisfy the equation
$$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$
Given that the line \(l\) is a tangent to \(E\),
- show that \(c ^ { 2 } = p m ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found.
The line \(l\) intersects the \(x\)-axis at the point \(A\) and intersects the \(y\)-axis at the point \(B\).
- Show that the area of triangle \(O A B\), where \(O\) is the origin, is
$$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
- Find the minimum area of triangle \(O A B\).
\hline
&
\hline
\end{tabular}