Conic sections are shapes formed by the intersection of a plane and a double-napped cone. The four types of conic sections are circles, parabolas, ellipses, and hyperbolas. A circle is defined as the set of all points equidistant from a fixed center point in a plane. The standard equation of a circle is (x-h)2 + (y-k)2 = r2, where (h,k) represents the center and r is the radius. This document provides examples of writing and graphing circle equations in standard and general form.

3. Conic Sections (Conics)
•Formed by intersecting
a double-napped cone
with a plane
•Circle
•Parabola
•Ellipse
•Hyperbola

4. Circle

5. Circle
•a conic section that is formed
by intersecting a cone with a
plane that is perpendicular to
the axis of the cone
•the set of all points in a plane
that are equidistant from a
fixed point called the center

6. Standard Form of the Equation of a Circle
(x−h)2
+ (y−k)2
= r2
where (h, k) is the center of the circle
r is the radius, r > 0

7. Example 1. Determine the standard equation of
the circle given the coordinates of its center and
the length of its radius
a. center at (2, -3) and r= 3
(x−h)2
+ (y−k)2
= r2
(x−2)2
+ [y−(−3)]2
= 32
(x−2)2
+ (y+3)2
= 9

8. Example 2. Given the standard form of the
equation below, find the coordinates of the center
and the radius.
a. (x+9)2
+ (y−1)2
= 25
Center is at (-9, 1) and r=5
(x+9)2
+ (y−1)2
= 25
[x− (−9)]2
+ (y−1)2
= 52

9. General Form of the Equation of a Circle
x2
+ y2
+ Dx + Ey + F = 0
Ex.
x2+ y2 + 3x + 12y + 2 = 0
x2
+ 2y2
-4x + 2y + 1 = 0

10. Transforming the Equation of a
Circle from the General Form
to the Standard Form and Vice
Versa

11. Example 3. Determine the standard form of
the equation of the circle defined by
4x𝟐
+ 4y𝟐
- 4x + 24y + 1 = 0
1
4
(4x2
+ 4y2
- 4x + 24y + 1 = 0)
4
4
x
2
+
4
4
y2
-
4
4
x +
24
4
y +
1
4
= 0
x2
+ y2
- x + 6y +
1
4
= 0
(x2
- x )+ (y2
+ 6y)= -
1
4
(x2
- x +
1
4
)+ (y2
+ 6y + 9)= -
1
4
+
1
4
+ 9
(x −
1
2
)
2
+ (y + 3)2
= 9

12. Example 4. Determine the general form of the
equation of the circle defined by
(x + 5)2
+ (y−6)2
= 4
(x + 5)2
+ (y−6)2
= 4
(𝑥2
+ 10x + 25) + (𝑦2
- 12y + 36) = 4
𝑥2
+ 𝑦2
+ 10x - 12y + 25 + 36 – 4 = 0
𝑥2+ 𝑦2+10x- 12y + 57= 0

13. Graph of a Circle
•The rectangular coordinate system
(Cartesian Coordinate Plane) is used to
sketch the graph of a circle. The graph
provides a clear view of its center and radius

14. Example 5. Write the equation for the graph
of the given circle below
Center is at (0,0), r= 2
(x−h)2
+ (y−k)2
= r2
(x−0)2
+ (y−0)2
= 22
𝑥2
+ y2
=4
x
y
0
1 2
-2 -1
-2
-1
1
2