1.
a) Let $x^{2}+y^{2}-a x+b y=c$ be the equation of a circle with centre $(2,3)$ and radius 4 . Find the values of $a, b$ and $c$.
\boxed{a = 4, b = -6, c = -3}
Step 1 :Given the equation of a circle with center (2, 3) and radius 4, we can write the equation as \((x - 2)^2 + (y - 3)^2 = 4^2\)
Step 2 :Expanding the equation, we get \(x^2 - 4x + y^2 - 6y - 3\)
Step 3 :Comparing the expanded equation with the given equation \(x^2 + y^2 - ax + by = c\), we find the values of a, b, and c
Step 4 :\boxed{a = 4, b = -6, c = -3}