Problem

The equation of a circle is $x^2+y^2+px+2y+q=0$, where $p$ and $q$ are constants.

(a) Express the equation in the form $(x-a)^2+(y-b)^2=r^2$, where $a$ is to be given in terms of $p$ and $r^2$ is to be given in terms of $p$ and $q$.

The line with equation $x+2y=10$ is the tangent to the circle at the point $A(4,3)$.

(b) (i) Find the equation of the normal to the circle at the point $A$.

(ii) Find the values of $p$ and $q$.