Determining the center and the radius of a circle

DETERMINING THE CENTER AND RADIUS OF A CIRCLE GIVEN ITS EQUATION AND VICE-VERSA

2. LEAST MASTERED SKILLS
Determining the Center and the Radius of a Circle
Given its Equation and Vice-Versa
LC CODE: M10GE-IIh-2
OBJECTIVES:
1. Identify the standard form of the equation
of a circle

2. Determine the center and the radius of a
circle given its equation and vice versa

3. Solve real-life problems using equation of a
circle

3. LEAST MASTERED SKILLS
Determining the Center and the Radius of a Circle
Given its Equation and Vice-Versa
LC CODE: M10GE-IIh-2
OBJECTIVES:
1. Identify the standard form of the equation
of a circle

2. Determine the center and the radius of a
circle given its equation and vice versa

3. Solve real-life problems using equation of a
circle

4. As you go through this lesson,
think of this important question:
Perform each activity to find the answer

“How does the equation of a circle facilitates in
finding solutions and making wise decision?”

5. Before you turn the next page,
try to answer the short quiz below...
1. Transform the equation into its standard form.
x
2
+y
2
+10x+4y-7=0
2. Determine the center and the radius of the following
equation.
1. x
2
+

y
2
=3
2

2. x+5)
2
+

(y+9)
2
=10
2

3. x
2
+y
2
+4x-4y-28=0

6. The standard equation of a circle with center at (h,k)
and a radius of r units is (x-h)
2
+

(y-k)
2
=r
2
.

7. If the center o the circle is at the
origin, the equation of the circle is
x
2
+

y
2
=r
2
.

8. The equation of a circle with center
at (1,3) and radius 5 is

(x-1)
2
+

(y-3)
2
=5
2


or

(x-1)
2
+

(y-3)
2
=25

9. The equation of a circle with center
at the origin and a radius of 3 is

x
2
+

y
2
=3
2


or

x
2
+

y
2
=9

10. The equation of a circle with center at
(-5, -9) and radius 10 is

(x+5)
2
+

(y+9)
2
=10
2


or

(x+5)
2
+

(y+9)
2
=100

11. The equation of a circle
with center at the origin
and a radius of 3 is

x
2
+

y
2
=3
2


or

x
2
+

y
2
=9

12. The equation of a circle with
center at
(0, -9) and radius 10 is

x
2
+

(y+9)
2
=10
2


or

x
2
+

(y+9)
2
=100

13. The equation of a circle with center at
(5, 0) and a radius of 4 is

(x-5)
2
+

y
2
=3
2


or

(x-5)
2
+

y
2
=9

14. Suppose two circles have the same center.
Should the equations defining these circles
be the same? Why?

15. The center and the radius of the
circle can be found given the
equation.

To do this, transform the equation
to its standard form. Remember
that the equation will be
(x-h)
2
+

(y-k)
2
=r
2
if the center is
(h, k), or x
2
+

y
2
=r
2
if the center of
the circle is at the origin.

16. Find the center and the radius of the
circle
x
2
+

y
2
=100.

Solution:
The equation x
2
+

y
2
=100 has its center
at the origin. Hence it can be trans-
formed to the form
x
2
+

y
2
= r
2
x
2
+

y
2
= 10
2

Then the center is at (0, 0) and its radius
is 10.

17. Determine the center and the radius of the
circle (x-5)
2
+

(y-8)
2
=5
2
.

The equation (x-5)
2
+

(y-8)
2
=5
2
can be written
in the form

(x-h)
2
+

(y-k)
2
=r
2


(x-5)
2
+

(y-8)
2
=5
2





(x-5)
2
+

(y-8)
2
=25

Then the center is at (5, 8) and the radius
is 5.

18. What is the center and the
radius of the circle
x
2
+y
2
-6x-10y+18=0?

The equation
x
2
+y
2
-6x-10y+18=0 is written
in general form.

x
2
+y
2
-6x-10y+18=0




x
2
-6x+y
2
-10y=-18
Add to both side of the equation:
½(-6)=-3; (-3)
2
=9
and
½(-10)=5; (-5)
2
= 25

Then
x
2
-6x+9+y
2
-10y+25=-18+9+25
(x
2
-6x+9)+(y
2
-10y+25)=16

Rewriting, we obtain
(x-3)
2
+(y-5)
2
=4
2


Therefore the center is at
(3, 5) and its radius is 4.

19. Write the standard form equation of each of the following circles
given the center and the radius.
Center Radius
1 (3, 8) 1
2 (-6, 4) 3
3 (9, -3) 5
4 (-1, -6) 7
5 (0, 0) 6
6 (0, 5) 4
7 (8, 0) 2

20. Transform the following equation to
standard form, then determine each
radius and center.
1. (x-2)
2
+(y-2)
2
-36=0
2. (x+4)
2
+(y-9)
2
-144=0
3. x
2
+y
2
-2x-8y-43=0
4. x
2
+y
2
+4x-4y-28=0
Question:
Is there a shorter way of transforming each equation
to standard form? Share your way.

21. Solve.

The diameter of the circle is 1 unit and its center
is at (-3, 8). What is the equation of the circle?
Write the equation in standard form.

22. I. Write the equation of the following
circles given the center and the radius.

Center Radius
1 (5, 9) 49
2 (-9, 12) 64
3 (8, -25) 121
4 (-3, -27) 36
5 (0, 0) 81
6 (0, -7) 169
7 (11, 0) 144

23. II. Find the center and the radius of
the following circles.
1. (x-7)
2
+(y+2)
2
=9
2. x
2
+(y+2)
2
=25
3. (x-5)
2
+y
2
=36
4. x
2
+y
2
=49

24. III. Transform the following equations in
standard form then determine the center
and the radius.
1. x
2
+y
2
+10x+4y-7=0
2. x2-y
2
-6x-8y-24=0

25. A radio signal can transmit messages up to a
distance of 3km. If the radio signal’s origin is located at a point
whose coordinates are (4,9), what is the equation of the circle
that defines the boundary up to which the messages can be
transmitted? Write the equation in standard form.

26. I. What defines me?
1. (x-3)
2+(y-8)
2=1
2
2. (x+6)
2+(y-4)
2=3
2
3. (x-9)
2+(y+3)
2=5
2
4. (x+1)
2+(y+6)
2=7
2
5. x
2+y
2=6
2
6. x
2+(y-5)
2=4
2
7. (x-8)
2+y
2=2
2
II. Find my Center
and Radius
1. (2, 2);6
2. (-4,9); 12
3. (2, 4); 8
4. (-2, 2); 6
III. Find Out More!
1. (x+3)
2+(y-8)
2=1
2

27. I.
1. (x-5)
2+(y-9)
2=7
2
2. (x+9)
2+(y-9)
2=8
2
3. (x-8)
2+(y+25)
2=11
2
4. (x+2)
2+(y+27)
2=6
2
5. x
2+y
2=9
2
6. x
2+(y+7)
2=13
2
7. (x-11)
2+y
2=12
2
II.
1. (7, -2);3
2. (0,-2); 5
3. (5, 0); 6
4. (0,0); 7
III.
1. (-5, -2); 6
2. (3, 4); 7
Enrichment
1. (x-5)
2+(y-9)
2=7
2

28. 1. Transform the equation into its standard form.
x
2
+y
2
+10x+4y-7=0
2. Determine the center and the radius of the following
equation.
1. x
2
+

y
2
=3
2

2. x+5)
2
+

(y+9)
2
=10
2

3. x
2
+y
2
+4x-4y-28=0

Let’s check your pre-test ...

29. 1.(x+5)
2
+(y+2)
2
=6
2


2. a. (0, 0);3
b. (-5, -9); 10
c. (2, -2); 6

30. http://tulyn.com/wordproblems/circle_equation.htm

https://www.wyckoffps.org/cms/lib/NJ01000588/Centricity/
Domain/161/11.3_Real-World%20Problems%20Circles.pdf

https://www.algebra.com/algebra/homework/word/
You are now
ready for the
next lesson ….