PROBLEMS ON EQUATIONS OF CIRCLES FOR SAT

Problem 1 :

A circle in he xy-plane is centered at (1, 2) and contains the point (4, 6). Which of the following could be the equation of the circle ?

a) (x - 1)² + (y - 2)² = 5           b) (x - 1)² + (y - 2)² = 25

c) (x + 1)² + (y + 2)² = 5          d) (x + 1)² + (y + 2)² = 25

Solution :

Equation of circle :

(x - h)² + (y - k)² = r²

Center (h, k) ==> (1, 2) and it passes through the point (4, 6).

(x - 1)² + (y - 2)² = r²  ----(1)

Put (x, y) as (4, 6)

(4 - 1)² + (6 - 2)² = r²

3² + 4² = r²

r² = 25

Applying the value of  r² in (1)

(x - 1)² + (y - 2)² = 25

So, option b is correct.

Problem 2 :

Which of the following is an equation of a circle in the xy-plane with center (3, -1) and a radius of 4 ?

a) (x - 3)² + (y + 1)² = 4      b) (x - 3)² + (y + 1)² = 16

c)  (x + 1)² + (y - 3)² = 4      d) (x + 3)² + (y - 1)² = 16

Solution :

Equation of circle :

(x - h)² + (y - k)² = r²

Here (h, k) is (3, -1) and radius (r) = 4

(x - 3)² + (y - (-1))² = 4²

(x - 3)² + (y + 1)² = 16

So, option b is correct.

Problem 3 :

(x + 6)² + (y - 4)² = 100

The equation above defines a circle in the xy-plane. The circle intercepts the y-axis at (0, 12) and (0, c). What is the value of c ?

Solution :

Applying the point (0, c) in the equation of circle, we get

(0 + 6)² + (c - 4)² = 100

6² + (c - 4)² = 100

(c - 4)² = 100 - 36

(c - 4)² = 64

(c - 4) = 8 and -8

c - 4 = 8 and c - 4 = -8

c = 8 + 4 and c = -8 + 4

c = 12 and c = -4

Problem 4 :

Which of the following is an equation of a circle in the xy-plane with center (2, -3) and a circumference of 20 π.

a) (x + 2)² + (y - 3)² = 20      b) (x - 2)² + (y + 3)² = 20

c)  (x - 2)² + (y + 3)² = 100      d) (x + 2)² + (y - 3)² = 400

Solution :

Circumference of a circle = 20 π

2πr = 20π

r = 10

radius of the circle = 10 and center (2, -3)

(x - 2)² + (y - (-3))² = r²

(x - 2)² + (y + 3)² = 10²

So, option c is correct.

Problem 5 :

x² + (y - 3)² = 25

The graph of the equation above in the xy-plane is a circle. At what points does the circle intersect the y-axis ?

a) (0, -2) and (0, 8)        b) (0, -3) and (0, 7)

c)  (0, -8) and (0, 2)        d)  (0, -22) and (0, 28)

Solution :

x² + (y - 3)² = 25

When the circle intersect the y-axis, then x = 0

0 + (y - 3)² = 25

(y - 3) = √25

y -.3 = 5 and -5

y - 3 = 5 and y - 3 = -5

y = 5+3 and y = -5+3

y = 8 and y = -2

So, y-intercepts are (0, 8) and (0, -2). option a is correct.

Problem 6 :

x² - 4x + y² + 6y = 12

The graph of the equation in the xy-plane is a circle. What is the circumference of the circle ?

a)  5π     b)  10π     c)  25π      d)  50π

Solution :

x² - 2 x (2) + 2² - 2² + y² + 2 y (3) + 3² - 3² = 12

(x - 2)² - 4 + (y + 3)² - 9 = 12

(x - 2)² + (y + 3)² - 13 = 12

(x - 2)² + (y + 3)²  = 12 + 13

(x - 2)² + (y + 3)²  = 25

(x - 2)² + (y + 3)²  = 5²

radius (r) = 5

Circumference of circle = 2πr

= 2π(5)

= 10π

So, the circumference of the circle is 10π.

Problem 7 :

A circle in the xy-plane passes through the point (2, 2) and has a radius of 5. Which of the following could be an equation of the circle ?

a) (x - 1)² + y²  = 5            b) (x + 2)² + (y - 5)²  = 25

c) (x - 2)² + (y - 2)²  = 25       d) (x - 7)² + (y - 7)²  = 25

Solution :

Equation of circle :

(x - h)² + (y - k)²  = r²

Option b :

(x + 2)² + (y - 5)²  = 25

radius = 5, let us check whether it passes through the point (2, 2).

(2 + 2)² + (2 - 5)²  = 25

4² + (-3)² = 25

16 + 9 = 25

25 = 25

Since option b satisfies the point, option b is correct.