Question

1. a _________ is the set of all points in a plane equidistant from a fixed point called the ______.
2. the distance from the center of a circle to any point on the circle is called the __________ and is often denoted by ( r ).
3. the standard form of an equation of a circle with center ( (h,k) ) and radius ( r ) is given by __________.
4. an equation of a circle written in the form ( x^{2}+y^{2}+ax + by + c = 0 ) is called the __________ form of an equation of a circle.

Explanation:

1. By the definition of a circle, it is the set of all points in a plane equidistant from a fixed - point (the center).
2. The radius is defined as the distance from the center of a circle to any point on the circle.
3. Using the distance formula between a general point \((x,y)\) and the center \((h,k)\) (which is equal to the radius \(r\)), we get \((x - h)^{2}+(y - k)^{2}=r^{2}\).
4. The form \(x^{2}+y^{2}+Ax + By + C = 0\) is called the general form of a circle's equation, which can be derived from expanding the standard form.

Answer:

1. circle, center
2. radius
3. \((x - h)^{2}+(y - k)^{2}=r^{2}\)
4. general