<s>[INST] Finding the equation of a circle given two points on the circle 
11. Find the equation of the circle which touches $x^{2} + y^{2} - 6x + 2y + 5 = 0$ at $(4, -3)$ and passes through $(0, 7)$.

My textbook has a worked example for obtaining the equation of a circle from three points on the circle. It also talks you through obtaining the equation of a circle if you're given two points on the circle and if it touches an axis (in this case you know a coordinate of the centre will be $\pm r$, where $r$ is your radius.)
I am reasonably well-practised at using these two techniques. I have also been practising finding the length of a tangent from a given point.
However, I have so far been unable to make the "jump" to this question, I suspect there's something I'm not seeing. So I was hoping for a hint that would help me work out how to approach this question.
 [/INST] I've come back to this question after a while and have found a solution which agrees with that in the textbook. My method is based primarily on the tips given by @Mufasa.
Let the circle which touches $x^{2} + y^{2} − 6x + 2y + 5 = 0$ at $(4,−3)$ and passes through $(0,7)$ be $C_{1}$.
Let the centre of $C_{1}$ be $(p, q)$ and let the radius be $a$.
$(4 - p)^{2} + (-3 - q)^{2} = a^{2}$
$p^{2} + (7 - q)^{2} = a^{2}$
Let the circle $x^{2} + y^{2} − 6x + 2y + 5 = 0$ be $C_{2}$.
The centre of $C_{2}$ is $(3, -1)$.
The gradient of the radius of $C_{2}$ to $(4, -3)$ is $-2$.
$\therefore$ The gradient of the tangent to $C_{2}$ at $(4, -3)$ is $\frac{1}{2}$.
$\therefore$ The gradient of the tangent to $C_{1}$ at $(4, -3)$ is $\frac{1}{2}$.
$\therefore$ The gradient of the radius of $C_{1}$ to $(4, -3)$ is $-2$.
$\therefore \dfrac{q - (-3)}{p - 4} = -2$
$2p + q - 5 = 0 \qquad (1)$
$(4 - p)^{2} + (-3 - q)^{2} = a^{2} \qquad (2)$
$p^{2} + (7 - q)^{2} = a^{2} \qquad (3)$
Equating (2) and (3), we ultimately get:
$p = \dfrac{5}{2}q - 3 \qquad (4)$
Substituting $p$ into (1), we get:
$q = \dfrac{11}{6}$
Substituting $q$ into (4), we get:
$p = \dfrac{19}{12}$
Substituting $p$ and $q$ into (3) we ultimately get:
$a^{2} = \dfrac{4,205}{144}$
This gives us our equation of $C_{1}$:
$$\left(x - \frac{19}{12}\right)^{2} + \left(y - \frac{11}{6}\right)^{2} = \frac{4,205}{144}$$
Expanding and simplifying, we get:
$$6x^{2} + 6y^{2} - 19x - 22y - 140 = 0.$$
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