Question

show that the four points (3,4),(-1,-4),(-1,2),(3,-6) are con-cyclic and the equation of the circle on which they lie??

Answer 1

The general form for the area of a circle is
\[(x-h)^2+(y-k)^2=r^2\]
You have x and y in each case, which means you have three variables. If you have three variables, you need at least 3 equations. We can generate them from the given points:
\[r^2=(3-h)^2+(4-k)^2\]\[r^2=(-1-h)^2+(-4-k)^2\]\[r^2=(-1-h)^2+(2-k)^2\]

Answer 2

Oh, there's an easy way...
-1 has two points connected to it, so the y-coordinate of the center is going to be the average of the y-coordinates for the two points. That gives us that the y-coordinate will be at -1.