Write down the equations of the vertical and horizontal asymptotes of the graph of \(f\).

The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).

Find the value of \(q\).

The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).

The point \({\text{P}}(x,{\text{ }}y)\) lies on the graph of \(f\). Show that \({\text{PQ}} = \sqrt {{{(x - 1)}^2} + {{\left( {\frac{3}{{x - 1}}} \right)}^2}} \).

The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).

Hence find the coordinates of the points on the graph of \(f\) that are closest to \((1,3)\).

\(x = q,{\text{ }}y = 3\)   (must be equations)     A1A1     N2

[2 marks]

recognizing connection between point of intersection and asymptote     (R1)

eg     \(x = 1\)

\(q = 1\)     A1     N2

[2 marks]

correct substitution into distance formula     A1

eg     \(\sqrt {{{(x - 1)}^2} + {{(y - 3)}^2}} \)

attempt to substitute \(y = \frac{{3x}}{{x - 1}}\)     (M1)

eg     \(\sqrt {{{(x - 1)}^2} + {{\left( {\frac{{3x}}{{x - 1}} - 3} \right)}^2}} \)

correct simplification of \(\left( {\frac{{3x}}{{x - 1}} - 3} \right)\)     (A1)

eg     \(\frac{{3x - 3x(x - 1)}}{{x - 1}}\)

correct expression clearly leading to the required answer     A1

eg     \(\frac{{3x - 3x + 3}}{{x - 1}},{\text{ }}\sqrt {{{(x - 1)}^2} + {{\left( {\frac{{3x - 3x + 3}}{{x - 1}}} \right)}^2}} \)

\({\text{PQ}} = \sqrt {{{(x - 1)}^2} + {{\left( {\frac{3}{{x - 1}}} \right)}^2}} \)     AG     N0

[4 marks]

recognizing that closest is when \({\text{PQ}}\) is a minimum     (R1)

eg     sketch of \({\text{PQ}}\), \(({\text{PQ}})'(x) = 0\)

\(x =  - 0.73205{\text{ }}x = 2.73205\)   (seen anywhere)     A1A1

attempt to find y-coordinates     (M1)

eg     \(f( - 0.73205)\)

\((-0.73205, 1.267949) , (2.73205, 4.73205)\)

\((-0.732, 1.27) , (2.73, 4.73) \)    A1A1     N4

[6 marks]