Find the coordinates of the centre of $C$ and its radius.

Write down the equation of $C$.

Find the coordinates of the second point on $C$ on the chord through $(1,{\text{ }}2)$ parallel to the tangent at $(3,{\text{ }}4)$.

METHOD 1

attempt to exploit the fact that the normal to a tangent passes through the centre $(a,{\text{ }}b)$     (M1)

EITHER

equation of normal is $y - 4 =  - \frac{1}{3}(x - 3)$     (A1)

obtain $a + 3b = 15$     A1

attempt to exploit the fact that a circle has a constant radius:     (M1)

obtain ${(1 - a)^2} + {(2 - b)^2} = {(3 - a)^2} + {(4 - b)^2}$     A1

leading to $a + b = 5$     A1

centre is $(0,{\text{ }}5)$     (M1)A1

radius $= \sqrt {{1^2} + {3^2}}  = \sqrt {10}$     A1

OR

gradient of normal $=  - \frac{1}{3}$     A1

general point on normal $= (3 - 3\lambda ,{\text{ }}4 + \lambda )$     (M1)A1

this point is equidistant from $(1,{\text{ }}2)$ and $(3,{\text{ }}4)$     M1

if $10{\lambda ^2} = {(2 - 3\lambda )^2} + {(2 + \lambda )^2}$

$10{\lambda ^2} = 4 - 12\lambda  + 9{\lambda ^2} + 4 + 4\lambda  + {\lambda ^2}$    A1

$\lambda  = 1$    A1

centre is $(0,{\text{ }}5)$     A1

radius $= \sqrt {10\lambda }  = \sqrt {10}$     A1

METHOD 2

attempt to substitute two points in the equation of a circle     (M1)

${(1 - h)^2} + {(2 - k)^2} = {r^2},{\text{ }}{(3 - h)^2} + {(4 - k)^2} = {r^2}$    A1

Note:     The A1 is for the two LHSs, which may be seen equated.

equate or subtract the equations

obtain $h + k = 5$ or equivalent     A1

attempt to differentiate the circle equation implicitly     (M1)

obtain $2(x - h) + 2(y - k)\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$     A1

Note:     Similarly, M1A1 if direct differentiation is used.

substitute $(3,{\text{ }}4)$ and gradient $= 3$ to obtain $h + 3k = 15$     A1

obtain centre $= (0,{\text{ }}5)$     (M1)A1

radius $= \sqrt {10}$     A1

[9 marks]

equation of circle is ${x^2} + {(y - 5)^2} = 10$     A1

[1 mark]

the equation of the chord is $3x - y = 1$     A1

attempt to solve the equation for the chord and that for the circle simultaneously     (M1)

for example ${x^2} + {(3x - 1 - 5)^2} = 10$     A1

coordinates of the second point are $\left( {\frac{{13}}{5},{\text{ }}\frac{{34}}{5}} \right)$     (M1)A1

[5 marks]

This question was usually well done, using a variety of valid approaches.

This question was usually well done, using a variety of valid approaches.

This question was usually well done, using a variety of valid approaches.