FIND CENTER AND RADIUS OF CIRCLE FROM STANDARD FORM

Problem 1 :

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 2 :

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 3 :

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 4 :

What is the circumference of the circle in the xy-plane with equation ?

(x + 7)² + (y - 5)² = 100 ?

a) 10π      b)  20π     c)  100π     d)  200π

Solution :

(x + 7)² + (y - 5)² = 100

(x + 7)² + (y - 5)² = 10²

Center of the circle is at (-7, 5)

Radius = 10

Circumference of circle = 2π r

= 2 π (10)

= 20π

Option b is correct.

Problem 5 :

Circle A has an area of 32π and circle B has an area of 384π. The radius of the circle B is how many times longer than the radius of circle A ?

a) 2√3     b)  3√2     c)  6    d) 12

Solution :

Let r₁ and r₂ be the radius of circles A and B.

Area of circle A = π r₁²

Area of circle B = π r₂²

π r₁² = 32π

r₁² = 32

r₁ = √32

r₁ = √(2 x 2 x 2 x 2 x 2)

= 4√2

π r₂² = 384π

r₂² = 384

r₂ = √384

r₂ = √(28 x 28)

= 8√6

8√6 = 4√2 (2√3)

So, radius of circle B is 2√3 times of circle A.

Problem 6 :

A circle in the xy-plane has the equation 

x² + y² = 9

The line y = 0 intersects this circle at two points. Which of the following is one of the points of intersection ?

a)  (-3, 0)         b) (0, -3)        c)  (0, 0)       d) (0, 3)

Solution :

x² + y² = 9

(x - 0)² + (y - 0)² = 3²

So, the point (-3, 0) is the point of intersection of the line y = 0.

Problem 7 :

Circle A in the xy-plane has the equation 

(x - 1)² + (y + 2)² = 64

Circle B is obtained by shifting circle A up 7 units, Which of the following equations represents circle B?

a) (x - 1)² + (y - 9)² = 64                b)  (x - 1)² + (y + 9)² = 64 

c) (x - 1)² + (y - 5)² = 64                d)  (x - 1)² + (y + 5)² = 64 

Solution :

(x - 1)² + (y + 2)² = 64

Moving up the circle 7 units, we get

(x - 1)² + (y - ((-2) + 7)² = 64

(x - 1)² + (y - 5)² = 64

Option c is correct.