Standard Equation

1. Circle…

4. The Standard Form of the Equation of a Circle
A. If the center is at the point of origin
c: (0,0)
𝑥2
+ 𝑦2
= 𝑟2
Ex: 𝑥2+𝑦2 = 52

5. The Standard Form of the Equation of a Circle
A. If the center is at (h,k)
c: (h,k)
(𝑥 − ℎ)2+(𝑦 − 𝑘)2= 𝑟2
Ex: (𝑥 − 2)2+(𝑦 − 3)2= 42

6. Example 1 :
A circle with center at (0,0)
and a radius of 7 units has
the equation
𝑥2
+ 𝑦2
= 72
or
𝑥2 + 𝑦2 = 49

7. Example 2:
The equation of a circle
with center at (2,7)
and a radius of 6 units is
(𝑥 − 2)2
+(𝑦 − 7)2
= 62
or
(𝑥 − 2)2+(𝑦 − 7)2= 36

8. Example 3:
The equation of a circle
with center at (-5,3)
and a radius of 4 units is
(𝑥 + 5)2
+(𝑦 − 3)2
= 42
or
(𝑥 + 5)2+(𝑦 − 3)2= 16

9. A circle with center C (h.k) and tangent to:
a. x- axis, the radius, r = /k/
b. y – axis , the radius, r = /h/
c. Line y = b, the radius r = /b-k/
d. Line x = a, the radius r = /a-h/
e. Line Ax+By+ C = 0, the radius r can be
computed using the distance formula
between a point and a line
𝑟 = 𝑑 =
/𝐴𝑥+𝐵𝑦+𝐶/
𝐴2+𝐵2

10. 1. Center (5,-6), and tangent to y-axis
Solution:
The center is 5 units away from the y-axis, so the
radius r is 5.
r=/h/ = /5/=5
(𝑥 − 5)2+(𝑦 + 6)2= 52
(𝑥 − 5)2
+(𝑦 + 6)2
= 25
Example 4: Determine the standard equation of the circle

11. 1. Center (5,-6), and tangent to x-axis
Solution:
The center is 6 units away from the y-axis, so the
radius r is 6.
r=/6/ = /6/=6
(𝑥 − 5)2+(𝑦 + 6)2= 62
(𝑥 − 5)2
+(𝑦 + 6)2
= 36
Example 5: Determine the standard equation of the circle

12. 1. Center (5,-6), and tangent to line y = -3
Solution:
r=/b-k/=/-3-(-6)/= 3
(𝑥 − 5)2
+(𝑦 + 6)2
= 32
(𝑥 − 5)2+(𝑦 + 6)2= 9
Example 6: Determine the standard equation of the circle

13. 1. Center (5,-6), and tangent to line x = -7
Solution:
r =/c-h/=/-7-5/= /-12/=12
(𝑥 − 5)2
+(𝑦 + 6)2
= 122
(𝑥 − 5)2+(𝑦 + 6)2= 144
Example 7: Determine the standard equation of the circle

14. 1. Center C(-1,-2), and Passes through the point
P(5,2)
r = CP = (𝑥𝑐 − 𝑥𝑝)2+(𝑦𝑐 − 𝑦𝑝)2
= (−1 − 5)2+(−2 − 2)2
= (−6)2+(−4)2
= 52
𝑟2
= 52
Example 8: Determine the standard equation of the circle

15. Question 1:
Determine the center and radius of the circle
𝑥2 + 𝑦2= 81
Center: (0,0)
Radius: 9

16. Question 2:
Determine the center and radius of the circle
(𝑥 − 5)2+(𝑦 + 4)2= 25
Center: (5,-4)
Radius: 5

17. Question 3:
Determine the center and radius of this graph:
Center: (-2, 4)
Radius: 7
Equation:
(𝑥 + 2)2 + 𝑦 − 4 2 = 49

18. Question 5:
Determine equation of the circle with
radius of 4 and center at origin
Answer: 𝑥2 + 𝑦2 = 16

19. Question 6:
Determine equation of the circle with
radius of 10 and center at (-4,-2)
Answer: (𝑥 + 4)2+(𝑦 + 2)2= 100

20. Question 7 :
Determine equation of the circle with
radius of 2 and center (-6,0)
Answer: (𝑥 + 6)2+ 𝑦2 = 4

21. Quiz: ½ crosswise
I. Write the equation of each of the following circles
given the center and the radius
1. Center: Origin Radius: 12 units
2. Center: (2,6) Radius: 9 units
3. Center: (-7,2) Radius: 15 units
II. Draw the graph of the following equation of circle:
4. 𝑥2
+ 𝑦2
= 100
5. (𝑥 + 7)2 + 𝑦 − 1 2 = 49

22. Quiz: ½ crosswise
III. Problem solving
Cellular phone networks use towers to transmit
calls to a circular area. On a grid of a province, the
coordinates that corresponds to the location of the
towers and the radius each covers are as follows: Wise
Tower is at (-5, -3) and covers 9 km radius; Global Tower
is at (3,6) and covers a 4 km radius; and Star Tower is at
(12, -3) and covers a 6 km radius.
Which tower transmits calls to phones located at:
1. (12,2) 2. (-6,-7) 3. (2,8)

23. The General Equation of a Circle
𝑥2
+ 𝑦2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
where D, E and F are real numbers

24. The General Equation of a Circle
Write the general equation of a circle with
center C(4,-1) and a radius of 7 units. Then
determine the values of D, E and F.
Solution:
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
(𝑥 − 4)2
+(𝑦 + 1)2
= 72