- The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$
where \(b\) is a constant and \(0 < b < 7\)
The eccentricity of the ellipse is \(e\)
- Write down, in terms of \(e\) only,
- the coordinates of the foci of \(E\)
- the equations of the directrices of \(E\)
Given that
- the point \(P ( x , y )\) lies on \(E\) where \(x > 0\)
- the point \(S\) is the focus of \(E\) on the positive \(x\)-axis
- the line \(l\) is the directrix of \(E\) which crosses the positive \(x\)-axis
- the point \(M\) lies on \(l\) such that the line through \(P\) and \(M\) is parallel to the \(x\)-axis
- determine an expression for
- \(P S ^ { 2 }\) in terms of \(e , x\) and \(y\)
- \(P M ^ { 2 }\) in terms of \(e\) and \(x\)
- Hence show that
$$b ^ { 2 } = 49 \left( 1 - e ^ { 2 } \right)$$
Given that \(E\) crosses the \(y\)-axis at the points with coordinates \(( 0 , \pm 4 \sqrt { 3 } )\)- determine the value of \(e\)
Given that the \(x\) coordinate of \(P\) is \(\frac { 7 } { 2 }\)
- determine the area of triangle \(O P M\), where \(O\) is the origin.