A circle is defined as all points that are equidistant from a fixed center point. The distance from the center point to any point on the circle is called the radius. The standard equation for a circle is x2 + y2 = r2, where the center is at the origin and r is the radius. This clearly shows the center and radius of the circle. More generally, the equation can be written as x2 + y2 + Dx + Ey + F = 0, where the values of D, E, and F depend on the coordinates of the center point and the radius.

1. CIRCLE
A circle is a set of all points (x,y) on the plane equidistant
from a fixed point C (h.k). The fixed point is called the
center of the circle, and the constant equal distance is called
the radius.

2. 𝑑 𝐶𝑃 = 𝑟
Let P(x,y) be a point on a circle. By definition, P moves so
that it is always a constant distance r from the fixed point
C(h,k), that is

3. (𝑥 − ℎ)2+(𝑦 − 𝑘)2 = r
Applying the distance formula,

4. (𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
Squaring both sides we have

5. If the center is at the origin,
then, reduces to
𝑥2
+ 𝑦2
=𝑟2
Which is the equation of the circle in the standard form. In
this form the equation clearly shows the center and the
radius that is why it is also known as the center radius
form.

6. 𝑥 2 + 𝑦 2 − 2hx − 2ky + (ℎ2 + 𝑘 2 − 𝑟 2 ) = 0
Expanding the parentheses in and collecting similar terms,
we get

7. 𝑥2
+ 𝑦2
+Dx+Ey+F = 0
If we let D = -2h, E = -2k, and F = ℎ2 + 𝑘2 − 𝑟2,
then the equation reduces to