Ellipse focus-directrix properties

8. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Determine the eccentricity of \(E\)
2. Hence, for this ellipse, determine
   1. the coordinates of the foci,
   2. the equations of the directrices. The point \(P\) lies on \(E\) and has coordinates \(( 3 \cos \theta , 2 \sin \theta )\). The line \(l _ { 1 }\) is the tangent to \(E\) at the point \(P\)
3. Using calculus, show that an equation for \(l _ { 1 }\) is $$2 x \cos \theta + 3 y \sin \theta = 6$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\) The line \(l _ { 1 }\) intersects the line \(l _ { 2 }\) at the point \(Q\)
4. Determine the coordinates of \(Q\)
5. Show that, as \(\theta\) varies, the point \(Q\) lies on the curve with equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = \alpha x ^ { 2 } + \beta y ^ { 2 }$$ where \(\alpha\) and \(\beta\) are constants to be determined.
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1. 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 \(e\)

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

2. The line with equation \(x = 9\) is a directrix of an ellipse with equation $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 8 } = 1$$ where \(a\) is a positive constant. Find the two possible exact values of the constant \(a\).

1. The line \(x = 8\) is a directrix of the ellipse with equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > 0 , b > 0$$ and the point \(( 2,0 )\) is the corresponding focus.
Find the value of \(a\) and the value of \(b\).

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 16 } = 1$$ The points \(S\) and \(S ^ { \prime }\) are the foci of \(E\).

1. Find the coordinates of \(S\) and \(S ^ { \prime }\)
2. Show that for any point \(P\) on \(E\), the triangle \(P S S ^ { \prime }\) has constant perimeter and determine its value.

An ellipse has equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\).

1. Sketch the ellipse. [1]
2. Find the value of the eccentricity \(e\). [2]
3. State the coordinates of the foci of the ellipse. [2]

An ellipse, with equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), has foci \(S\) and \(S'\).

1. Find the coordinates of the foci of the ellipse. [4]
2. Using the focus-directrix property of the ellipse, show that, for any point \(P\) on the ellipse, $$SP + S'P = 6.$$ [3]