Show that the area enclosed by the ellipse is \(\pi ab\).

Determine which coordinate axis the major axis of the ellipse lies along.

Hence find the eccentricity.

Find the coordinates of the foci.

Find the equations of the directrices.

The centre of another ellipse is now given as the point (2, 1). The minor and major axes are of lengths 3 and 5 and are parallel to the \(x\) and \(y\) axes respectively. Find the equation of the ellipse.

\(A = 4\int {y{\text{d}}x} \)      (M1)

\(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \Rightarrow \)

\(y = \frac{{b\sqrt {{a^2} - {x^2}} }}{a}\)      (A1)

let \(x = a\,{\text{cos}}\,\theta  \Rightarrow y = b\,{\text{sin}}\,\theta \)      M1

\(\frac{{{\text{d}}x}}{{{\text{d}}\theta }} =  - a\,{\text{sin}}\,\theta \)      A1

when \(x = 0,\,\,\theta  = \frac{\pi }{2}\). When \(x = a,\,\,\theta  = 0\)       A1

\( \Rightarrow A = 4\int_{\frac{\pi }{2}}^0 {b\,{\text{sin}}\,\theta } \left( { - a\,{\text{sin}}\,\theta } \right){\text{d}}\theta \)      M1

\( \Rightarrow A =  - 4ab\int_{\frac{\pi }{2}}^0 {\,{\text{si}}{{\text{n}}^2}\,\theta } \,{\text{d}}\theta \)

\( \Rightarrow A =  - 2ab\int_{\frac{\pi }{2}}^0 {\,\left( {1 - \,{\text{cos}}\,2\theta } \right)} \,{\text{d}}\theta \)      M1

\( \Rightarrow A =  - 2ab\left[ {\theta  - \frac{{{\text{sin}}\,2\theta }}{2}} \right]_{\frac{\pi }{2}}^0\)     A1

\( \Rightarrow A =  - 2ab\left[ {0 - 0 - \left( {\frac{\pi }{2} - 0} \right)} \right]\)     M1

\( \Rightarrow A = \pi ab\)      AG

[9 marks]

\(b = 2\)

hence \(2\pi a = 8\pi  \Rightarrow a = 4\)      A1

hence major axis lies along the x-axis      A1

[2 marks]

\({b^2} = {a^2}\left( {1 - {e^2}} \right)\)      (M1)

\(4 = 16\left( {1 - {e^2}} \right) \Rightarrow e = \frac{{\sqrt 3 }}{2}\)      A1

[2 marks]

coordinates of foci are \(\left( { \pm ae,\,0} \right) = \left( {2\sqrt 3 ,\,0} \right),\,\left( { - 2\sqrt 3 ,\,0} \right)\)      A1A1

[2 marks]

equations of directrices are \(x =  \pm \frac{a}{e} = \frac{8}{{\sqrt 3 }},\, - \frac{8}{{\sqrt 3 }}\)      A1A1

[2 marks]

\(a = \frac{3}{2},\,b = \frac{5}{2}\)     (A1)

hence equation is \(\frac{4}{9}{\left( {x - 2} \right)^2} + \frac{4}{{25}}{\left( {y - 1} \right)^2} = 1\)     M1A1

[3 marks]