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]