(i)     Find the equation of the tangent to the ellipse at the point \(\left( {1,{\text{ }}\frac{1}{{\sqrt 3 }}} \right)\).

(ii)     Find the equation of the normal to the ellipse at the point \(\left( {1,{\text{ }}\frac{1}{{\sqrt 3 }}} \right)\).

Given that the tangent crosses the \(x\)-axis at P and the normal crosses the \(y\)-axis at Q, find the equation of (PQ).

Hence show that (PQ) touches the ellipse.

State the coordinates of the point where (PQ) touches the ellipse.

Find the coordinates of the foci of the ellipse.

Find the equations of the directrices of the ellipse.

(i)     \(2x + 6y\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\)     M1

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} =  - \frac{x}{{3y}}\)     A1

gradient of tangent is \( - \frac{{\sqrt 3 }}{3}\)     A1

equation of tangent is \(y - \frac{1}{{\sqrt 3 }} =  - \frac{{\sqrt 3 }}{3}(x - 1)\)     M1A1

\(\left( { \Rightarrow y =  - \frac{{\sqrt 3 }}{3}x + \frac{{\sqrt 3 }}{3} + \frac{1}{{\sqrt 3 }} \Rightarrow y =  - \frac{{\sqrt 3 }}{3}x + \frac{2}{{\sqrt 3 }}} \right)\)

(ii)     gradient of normal is \(\sqrt 3 \)     A1

equation of normal is \(y - \frac{1}{{\sqrt 3 }} = \sqrt 3 (x - 1)\)     A1

\(\left( { \Rightarrow y = x\sqrt 3  - \sqrt 3  + \frac{1}{{\sqrt 3 }} \Rightarrow y = \sqrt 3 x - \frac{2}{{\sqrt 3 }}} \right)\)

coordinates of P are \((2,{\text{ }}0)\)     A1

coordinates of Q are \(\left( {0,{\text{ }} - \frac{2}{{\sqrt 3 }}} \right)\)     A1

equation of (PQ) is \(\frac{{y - 0}}{{x - 2}} = \frac{{\frac{2}{{\sqrt 3 }}}}{2}\)     M1

\( \Rightarrow y = \frac{1}{{\sqrt 3 }}(x - 2)\)     A1

substitute equation of (PQ) into equation of ellipse

\({x^2} + 3{\left( {\frac{{x - 2}}{{\sqrt 3 }}} \right)^2} = 2\)     M1A1

\( \Rightarrow {x^2} + {x^2} - 4x + 4 = 2\)

\( \Rightarrow {(x - 1)^2} = 0\)     A1

since the equation has two equal roots (PQ) touches the ellipse     R1

\(\left( {1,{\text{ }} - \frac{1}{{\sqrt 3 }}} \right)\)     A1

\({x^2} + 3{y^2} = 2\)

\(\frac{{{x^2}}}{2} + \frac{{{y^2}}}{{\frac{2}{3}}} = 1\)

\( \Rightarrow a = \sqrt 2 ,{\text{ }}b = \sqrt {\frac{2}{3}} \)     A1

EITHER

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

\(\frac{2}{3} = 2(1 - {e^2})\)     M1

\( \Rightarrow e = \sqrt {\frac{2}{3}} \)     A1

coordinates of foci are \(( \pm ae,{\text{ }}0) \Rightarrow \left( {\frac{2}{{\sqrt 3 }},{\text{ }}0} \right),{\text{ }}\left( { - \frac{2}{{\sqrt 3 }},{\text{ }}0} \right)\)     A1

OR

\({f^2} = {a^2} - {b^2}\)     M1

\({f^2} = 4 - \frac{2}{3}\)     A1

coordinates of foci are \(\left( {\frac{2}{{\sqrt 3 }},{\text{ }}0} \right),{\text{ }}\left( { - \frac{2}{{\sqrt 3 }},{\text{ }}0} \right)\)     A1

Note: Award accuracy marks if \({a^2}\), \({b^2}\) and \({e^2}\) are given.

EITHER

equations of directrices are \(x =  \pm \frac{a}{e} \Rightarrow x = \sqrt 3 ,{\text{ }}x =  - \sqrt 3 \)     A1

OR

\(d = \frac{{{a^2}}}{f} \Rightarrow x = \sqrt 3 ,{\text{ }}x =  - \sqrt 3 \)     A1

Parts a) and b) were well done by most candidates, but surprisingly many candidates lost marks on part c). Parts e) and f) were only completed successfully by a small number of candidates and it was common to see parts a) and b) fully correct, parts c) and d) attempted but not fully correct and parts e) and f) not attempted at all.

Parts a) and b) were well done by most candidates, but surprisingly many candidates lost marks on part c). Parts e) and f) were only completed successfully by a small number of candidates and it was common to see parts a) and b) fully correct, parts c) and d) attempted but not fully correct and parts e) and f) not attempted at all.

Parts a) and b) were well done by most candidates, but surprisingly many candidates lost marks on part c). Parts e) and f) were only completed successfully by a small number of candidates and it was common to see parts a) and b) fully correct, parts c) and d) attempted but not fully correct and parts e) and f) not attempted at all.

Parts a) and b) were well done by most candidates, but surprisingly many candidates lost marks on part c). Parts e) and f) were only completed successfully by a small number of candidates and it was common to see parts a) and b) fully correct, parts c) and d) attempted but not fully correct and parts e) and f) not attempted at all.

Parts a) and b) were well done by most candidates, but surprisingly many candidates lost marks on part c). Parts e) and f) were only completed successfully by a small number of candidates and it was common to see parts a) and b) fully correct, parts c) and d) attempted but not fully correct and parts e) and f) not attempted at all.

Parts a) and b) were well done by most candidates, but surprisingly many candidates lost marks on part c). Parts e) and f) were only completed successfully by a small number of candidates and it was common to see parts a) and b) fully correct, parts c) and d) attempted but not fully correct and parts e) and f) not attempted at all.