equations of circle IN STANDARD AND GENERAL FORM

1. THE CIRCLE
LESSON 6

2. CONTENTS
The following will be discussed in this presentation:
◦Chapter 6 of Analytic Geometry by Love and Rainville
Definitions; General Form
Equation of a Circle
Reduction of the General Form to the Center-Radius
Form
Circle Determined by Three Conditions
Tangents to a Circle

3. DEFINITION; GENERAL FORM
A circle is the locus of a point that moves at a
constant distance from a fixed point. The fixed point
is called the Center and the constant distance is the
Radius. The circle is one of the curves represented by
an equation of 2nd Degree in the form:
𝐴𝑥2
+ 𝐵𝑥𝑦 + 𝐶𝑦2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0

4. DEFINITION; GENERAL FORM
A circle is mostly conveniently represented in the
general form
𝐴𝑥2
+ 𝐶𝑦2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
Letting A=C and B=0.
**Theorem: An equation of the second degree in
which 𝑥2
and 𝑦2
have equal coefficient and the xy-
term is missing represents a circle (exceptionally, a
single point or no locus)

5. EQUATION OF A CIRCLE
oCircle with Center at the Origin
oCircle with Center (h, k)

6.  Circle with Center at the Origin
𝑑 = 𝑟
𝑟 = 𝑥 − 0 2 + 𝑦 − 0 2
𝑟 = 𝑥2 + 𝑦2
2
𝑟2 = 𝑥2 + 𝑦2
Thus, the standard equation of a
circle with center at the origin is
𝑥2 + 𝑦2 = 𝑟2
P(x, y)
O
y
x
d

7.  Circle with Center at (h, k)
𝑑 = 𝑟
𝑟 = 𝑥 − ℎ 2 + 𝑦 − 𝑘 2
𝑟 = 𝑥 − ℎ 2 + 𝑦 − 𝑘 2
2
𝑟2 = 𝑥 − ℎ 2 + 𝑦 − 𝑘 2
Thus, the standard equation of a
circle with center at the origin is
𝑥 − ℎ 2 + 𝑦 − 𝑘 2 = 𝑟2
P(x, y)
C(h, k)
h
k
x
y
O
d

8. EQUATION OF A CIRCLE
Example 1:
Write the equation of the circle with center at (2, -3) and
radius 5. Sketch the graph.

9. EQUATION OF A CIRCLE
Example 2:
Write the equation of the circle with center at the origin
and radius 3. Sketch the graph.

10. EQUATION OF A CIRCLE
Example 3:
Write the equation of the circle with the points (2, 5) and
(6, -1) as the endpoints of its diameter. Sketch the graph.

11. EQUATION OF A CIRCLE
Example 4:
Write the equation of the circle with center at the (1,1)
and touching the line 3x + 4y = 10. Sketch the graph.

12. REDUCTION OF THE GENERAL
FORM TO CENTER-RADIUS FORM
GENERAL FORM:
𝐴𝑥2
+ 𝐶𝑦2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
CENTER-RADIUS FORM:
𝑥 − ℎ 2
+ 𝑦 − 𝑘 2
= 𝑟2

13. REDUCTION OF THE GENERAL
FORM TO CENTER-RADIUS FORM
Example 1:
Find the center and radius of the circle represented by
the equation 4𝑥2
+ 4𝑦2
− 4𝑥 + 2𝑦 + 1 = 0

14. REDUCTION OF THE GENERAL
FORM TO CENTER-RADIUS FORM
Example 2:
Show that the circles 𝑥2
+ 𝑦2
− 2𝑥 − 3 = 0 and 𝑥2
+
𝑦2
+ 4𝑥 − 8𝑦 + 11 = 0 are tangent to each other.
Sketch the graph

15. REDUCTION OF THE GENERAL
FORM TO CENTER-RADIUS FORM
Example 3:
Find the points of intersection of the circle
𝑥2
+ 𝑦2
− 18𝑥 − 4𝑦 + 35 = 0 and
𝑥2 + 𝑦2 + 2𝑥 + 6𝑦 − 15 = 0. Sketch the
graph

16. CIRCLE DETERMINED BY THREE
CONDITIONS
Example 1:
Find the equation of the circle through the points
P1:(1,1), P2:(2, -1), and P3:(2, 3).

17. TANGENTS TO A CIRCLE
Line Tangent to a Circle – a line that touches the circle
at exactly one point.
x
y
O
TANGENT LINE

18. TANGENTS TO A CIRCLE
Example 1:
Find the equation of the tangent line at (-1, 4) on the
circle 𝑥2
+ 𝑦2
− 4𝑥 − 21 = 0.