Recall the:
Distance Formula:
The distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ is $ D=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $
Example:
The radius of your circle is the distance between the points $(-1,4)$ and $(3,-2)$. Using the Distance Formula: $ D= \sqrt{ \bigl(3-(-1)\bigr)^2 + (-2-4)^2 } = \sqrt{ 4^2 + (-6)^2 } = \sqrt{ 16+36 } = \sqrt{ 52 } . $
What is the equation of the circle?
It is important to realize the "equation of the circle" is: a point $(x,y)$ is on the circle if and only if the coordinates of the point $x$ and $y$ satisfy the equation. So, how to get the equation? What is the relationship between the $x$ and $y$ coordinates of a point on the circle?
Well, let $(x,y)$ be a point on the circle. The big idea is:
$ \text{The distance from the point }(x,y)\text{ to the center }(-1,4)\text{ is }\sqrt{52}.$
So, using the distance formula (with $(x_2,y_2)=(x,y)$ and $(x_1,y_1)=(-1,4)$) , it follows that $ \sqrt{52}=\sqrt{\bigl(x-(-1)\bigr)^2 +(y-4)^2}. $
Or $ 52=(x+1)^2 +(y-4)^2. $
The shortcut would be to just use the following formula (But it's important to realize why you'd use it and where it comes from):
Equation of a Circle
The equation of the circle with center located at $(a,b)$ and with radius $r$ is $ r^2=(x-a)^2 +(y-b)^2 $ Note that the radius squared is on the left-hand side of the equation.
The following may help: