Pre-Calculus

1. CONIC
SECTIONS
-CIRCLES-

2. CIRCLES
• Geometric: the result of a
cone and a plane
intersecting each other
• Algebraic: set of points on a
plane that are equidistant
from a fixed point on the
same plane (center/origin)

3. x
y
Radius (r)
Center (h,k)
- The distance from the center of the circle (h,k) to any point on the
circle (x,y).
RADIUS
(x,y)

4. Equation of the Circle
• The center-radius or standard form:
(𝑥 − ℎ)2+(𝑦 − 𝑘)2 = 𝑟2
where (h,k) is the center and r is the radius
• Derived from the Pythagorean Theorem
𝑎2 + 𝑏2 = 𝑟2 𝑎
𝑏

5. Find the center and the radius for each of the
following circles.
1. (𝑥 − 4)2
+(𝑦 − 6)2
= 25
2. (𝑥 − 3)2
+(𝑦 + 1)2
= 16
3. (𝑥 + 5)2
+ 𝑦 − 2 = 15
4. 𝑥2
+ (𝑦 + 7)2
= 18
ans. center is (4,6) & r = 5
ans. center is (3,-1) & r = 4
ans. center is (-5,2) & r = 15
ans. center is (0,-7) & r = 3 2

6. Write the equation of the circle given the
center and the radius.
1. C (2,-7) & r = 3
(𝑥 − 2)2+(𝑦 + 7)2= 9
2. C (-3, -3) & r = 10
(𝑥 + 3)2+(𝑥 + 3)2= 10

7. Write the equation of a circle whose diameter
is the line segment joining A(-3,-4) and B(4,3).
• What must you find first?
The center and the radius.
• How can you find the center?
The center is the midpoint of the segment.
(½ , - ½ )
• How can you find the radius?
The radius is the distance from the center to a point on the circle.
Use the distance formula.
r = 7
1
2
2 2
1 1 49
2 2 2
x y
   
      
   

8. Find the equation of the circle.
𝑥2
+ 𝑦2
+ 6𝑥 − 4𝑦 − 12 = 0
Hint: You must complete the square
2 2
6 ___ 4 ___ 12x x y y     
2 2
6 9 4 4 12 9 4x x y y       
2 2
( 3) ( 2) 25x y   