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.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM