FIND CENTER AND RADIUS OF CIRCLE FROM EQUATION OF A CIRCLE

Problem 1 :

(x−8)² + (y−6)² = 36

For the equation above, what is the coordinate point for the center of the circle as well as the circle’s radius?

A) Center: (-8, -6), Radius: 36

B) Center: (8, 6), Radius: 36

C) Center: (-8, -6), Radius: 6

D) Center: (8, 6), Radius: 6

Solution :

Comparing the given equation with 

(x−8)² + (y−6)² = 6²

(x−h)² + (y−k)² = r²

(h, k) is (8, 6) and radius = 6

So, option D is correct.

Problem 2 :

(x+14)² + (y+2)² = 64

For the equation above, what is the coordinate point for the center of the circle as well as the circle’s radius?

A) Center: (14, 2), Radius: 8

B) Center: (14, 2), Radius: 64

C) Center: (-14, -2), Radius: 8

D) Center: (-14, -2), Radius: 64

Solution :

Comparing the given equation with 

(x+14)² + (y+2)² = 64

(x-(-14))² + (y-(-2))² = 8²

(x−h)² + (y−k)² = r²

(h, k) is (-14, -2) and radius = 8

So, option C is correct.

Problem 3 :

(x−11)² + (y+4)² = 49

For the equation above, what is the coordinate point for the center of the circle as well as the circle’s radius?

A) Center: (11, -4), Radius: 7

B) Center: (11, -4), Radius: 49

C) Center: (-11, 4), Radius: 49

D) Center: (-11, 4), Radius: 7

Solution :

Comparing the given equation with 

(x-11)² + (y+4)² = 49

(x-11)² + (y-(-4))² = 7²

(x−h)² + (y−k)² = r²

(h, k) is (11, -4) and radius = 7

So, option A is correct.

Problem 4 :

x² + 10x + y² – 6y = -18

The graph of the equation shown above is a circle. What is the radius of the circle?

A) 3    B) 4     C) 5    D) 9

Solution :

x² + 10x + y² – 6y = -18

x² + 2⋅x⋅5 + y² – 2⋅y⋅3 = -18

To complete the formula

x² + 2⋅x⋅5 + 5² - 5² + y² – 2⋅y⋅3 + 3² - 3² = -18

(x + 5)² + (y - 3)² - 25 - 9 = -18

(x + 5)² + (y - 3)² = -18 + 25 + 9

(x + 5)² + (y - 3)² = 16

(x + 5)² + (y - 3)² = 4²

(x - (-5))² + (y - 3)² = 4²

Comparing with

(x−h)² + (y−k)² = r²

(h, k) is (-5, 3) and radius = 4

So, option B is correct.

Problem 5 :

x² + 18x + y² – 8y = -48

The graph of the equation shown above is a circle. What is the radius of the circle?

A) 4    B) 5    C) 6     D) 7

Solution :

x² + 18x + y² – 8y = -48

x² + 2⋅x⋅9 + y² – 2⋅y⋅4 = -48

To complete the formula

x² + 2⋅x⋅9 + 9² - 9²+ y² – 2⋅y⋅4 + 4² - 4² = -48

(x + 9)² + (y - 4)² - 81 - 16 = -48

(x + 9)² + (y - 4)² - 97 = -48

(x - (-9))² + (y - 4)² = -48 + 97

(x - (-9))² + (y - 4)² = 49

(x - (-9))² + (y - 4)² = 7²

Radius = 7

Problem 6 :

x² - 4x + y² + 6y = 113

The graph of the equation shown above is a circle. What is the coordinate point of the center of the circle?

A) (13, 10)    B) (4, 13)     C) (-4, 6)     D) (2, -3)

Solution :

x² - 4x + y² + 6y = 113

x² - 2⋅x⋅2 + y² + 2⋅y⋅3 = 113

x² - 2⋅x⋅2 + 2² - 2² + y² + 2⋅y⋅3 + 3² - 3² = 113

(x - 2)² - 4 + (y + 3)² - 9 = 113

(x - 2)² + (y + 3)² - 13 = 113

(x - 2)² + (y + 3)² = 113 + 13

(x - 2)² + (y + 3)² = 126

h = 2 and k = -3

So, the center of the circle is (2, -3),

Problem 7 :

Find the radius of a circle that has equation (x - 2)²+ (y - 3)² = r² and contains (2, 5)

Solution :

(x - 2)²+ (y - 3)² = r²

The circle passes through the point (2, 5),

(2 - 2)²+ (5 - 3)² = r²

0²+ 2² = r²

r = 2

So, the radius of the circle is 2 units.

Problem 8 :

A pizza parlor is located at the coordinates (7, 3) on a coordinate grid. The pizza parlor’s delivery service extends for 15 miles. Write the equation of the circle which represents the outer edge of the pizza delivery service area.

Solution :

From the given information, it is clear that the center of the circle is (7, 3).

Radius = 15 miles

(x - h)²+ (y - k)² = r²

(x - 7)²+ (y - 3)² = 15²

(x - 7)²+ (y - 3)² = 225

Problem 9 :

The segment with endpoints A(1, -2) and B(1, 6) is the diameter of a circle.

a. Graph the points and draw the circle.

b. What is the circumference of the circle?

c. What is the equation of the circle?

Solution :

a)

Midpoint of the diameter = (x₁ + x₂)/2, (y₁ + y₂)/2

= (1 + 1)/2, (-2 + 6)/2

= 2/2, 4/2

= (1, 2)

Distance between center and one end point of the diameter

= √(x₂ - x₁)² + (y₂ - y₁)²

= √(1 - 1)² + (2 + 2)²

= √0² + 4²

radius = 4

b) Circumference of circle = 2 πr

= 2  π (4)

= 8 π

c) Equation of circle :

(x - h)²+ (y - k)² = r²

(x - 1)²+ (y - 2)² = 4²

(x - 1)²+ (y - 2)² = 16