DETERMINE IF A POINT LIES INSIDE OR OUTSIDE OF A CIRCLE

Problem 1 :

Determine whether the points (2, 1), (7, -3) and (3, -4) lie within, out with or on the circumference of the circle.

Solution :

Point (2, 1)

(x – 2)² + (y + 3)²

= (2 - 2)² + (1 + 5)²

= (0)² + (6)²

= 36 > 29

The point lies outside the circle.

Point (7, -3)

(x – 2)² + (y + 3)²

= (7 - 2)² + (-3 + 5)²

= 5² + 2²

= 25 + 4

= 29

The point lies on the circle.

Point (3, -4)

(x – 2)² + (y + 3)²

= (3 - 2)² + (-4 + 5)²

= 1² + 1²

= 2 < 29

The point lies inside the circle.

Problem 2 :

State : “ Yes” or “ No”. Does the point (-3, 45) lie on the circle centered at

(-6, -5), and containing the point (15, -35)?

Solution :

Center = (-6, -5) 

Point = (15, -35)

x₁ = 15, x₂ = -6, y₁ = -35, y₂ = -5

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √(-6 – 15)² + (-5 + 35)²

= √((-21)² + (30)²)

= √(441 + 900)

= √1341

= 36.62

Center = (-6, -5) 

Point = (-3, 45)

x₁ = -3, x₂ = -6, y₁ = 45, y₂ = -5

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √(-6 – (-3))² + (-5 - 45)²

= √((-3)² + (-50)²)

= √(9 + 2500)

= √2509

= 50.09

36.62 < 50.09

So, the point lies outside the circle.

Problem 3 :

State : “ Yes” or “ No”. Does the point (5, -28) lie on the circle centered at

(-2, -4), and containing the point (-9, 20)?

Solution :

Center = (-2, -4) 

Point = (5, -28)

x₁ = 5, x₂ = -2, y₁ = -28, y₂ = -4

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √(-2 – 5)² + (-4 + 28)²

= √((-7)² + (24)²)

= √(49 + 576)

= √625

= 25  ----(1)

Center = (-2, -4) 

Point = (-9, 20)

x₁ = -9, x₂ = -2, y₁ = 20, y₂ = -4

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √(-2 – (-9))² + (-4 - 20)²

= √(7)² + (-24)²

= √(49 + 576)

= √625

= 25   ----(2)

25 = 25

 So, the point lies on the circle.

Problem 3 :

State : “ Yes” or “ No”. Does the point (3, 7) lie on the circle centered at

(2, 2), and containing the point (6, 3)?

Solution :

Center = (2, 2) 

Point = (3, 7)

x₁ = 3, x₂ = 2, y₁ = 7, y₂ = 2

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √(2 – 3)² + (2 - 7)²

= √((-1)² + (-5)²)

= √(1 + 25)

= √26

= 5.10

Center = (2, 2) 

Point = (6, 3)

x₁ = 6, x₂ = 2, y₁ = 3, y₂ = 2

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √(2 – 6)² + (2 - 3)²

= √(-4)² + (-1)²

= √(16 + 1)

= √17

= 4.12

5.10 > 4.12

 So, the point lies outside the circle.

Problem 4 :

State : “ Yes” or “ No”. Does the point (-6, -52) lie on the circle centered at

(-5, -8), and containing the point (16, 68)?

Solution :

Center = (-5, -8) 

Point = (-6, -52)

x₁ = -6, x₂ = -5, y₁ = -52, y₂ = -8

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √((-5) + 6)² + ((-8) + 52))²

= √(1)² + (44)²)

= √(1 + 1936)

= √1937

= 44.01

Center = (-5, -8) 

Point = (16, 68)

x₁ = 16, x₂ = -5, y₁ = 68, y₂ = -8

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √((-5) – (16))² + ((-8) - 68)²

= √(-21)² + (-76)²

= √(441 + 5776)

= √6217

= 78.85

So, the point lies inside the circle.

Problem 5 :

State : “ Yes” or “ No”. Does the point (-4, -1) lie on the circle centered at

(3, 5), and containing the point (7, 2)?

Solution :

Center = (3, 5) 

Point = (-4, -1)

x₁ = -4, x₂ = 3, y₁ = -1, y₂ = 5

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √(3 – (-4))² + (5 – (-1))²

= √(7)² + (6)²)

= √49 + 36)

= √85

= 9.22

Center = (3, 5) 

Point = (7, 2)

x₁ = 7, x₂ = 3, y₁ = 2, y₂ = 5

R = √(x₂ – x₁)² + (y₂ – y₁)²

= √(3 – 7)² + (5 - 2)²

= √(-4)² + 3)²

= √(16 + 9)

= √25

= 5

So, the point lies outside the circle.