# DETERMINE IF A POINT LIES INSIDE OR OUTSIDE OF A CIRCLE

Given a circle with center (a, b) and radius r units, we can determine whether a point (p, q) lies within, out with or on the circumference using the following rules :

(p – a)2 + (q – b)2 < r2 = the point lies within the circle

(p – a)2 + (q – b)2 = r2 = the point lies on the circumference of the circle

(p – a)2 + (q – b)2 > r2 = the point lies out with the 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.

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