section 2.4 – circles
DESCRIPTION
Section 2.4 – Circles. Circle – a set of points in a plane that are equidistant from a fixed point. Section 2.4 – Circles. Definitions. Radius (r) – is the distance that the set of points in a circle are from the fixed. Center (h, k) – the coordinates of the fixed point. r. (h , k ). - PowerPoint PPT PresentationTRANSCRIPT
Section 2.4 – CirclesCircle – a set of points in a plane that are equidistant from a fixed point.
Section 2.4 – Circles
Radius (r) – is the distance that the set of points in a circle are from the fixed.Center (h, k) – the coordinates of the fixed point.
Definitions
r(h, k)
Section 2.4 – Circles
(x, y)
(0, 0)
r
Section 2.4 – Circles
Section 2.4 – Circles
Section 2.4 – Circles
x-intercepts: set the y variable equal to zero and solve for x.
y-intercepts: set the x variable equal to zero and solve for y.
Intercepts
(𝑥−h)2+(0−𝑘)2=𝑟2
(0−h)2+(𝑦−𝑘)2=𝑟2
(0 , 𝑦1)
(0 , 𝑦2)
(𝑥1 ,0) (𝑥2 ,0)
Section 2.4 – Circles
𝑥2+ 𝑦2=36Find the coordinates of the center, the radius and the intercepts.Examples (𝑥−h)2+(𝑦−𝑘)2=𝑟2
𝑟2=36
𝐶𝑒𝑛𝑡𝑒𝑟
𝑟=6𝑅𝑎𝑑𝑖𝑢𝑠
𝑥−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 (𝑦=0)𝑥2+02=36 𝑥=±6(−6,0) (6,0)𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 (𝑥=0)02+𝑦2=36𝑦=±6(0 ,−6) (0,6 )
(−6,0)
(0 ,−6)
(0,0)𝑟=6
(0,6 )
(6,0)
(0,0)
Section 2.4 – Circles
Find the coordinates of the center, the radius and the intercepts.Examples
(5 ,−4)
(𝑥−5)2+(𝑦+4 )2=41
(𝑥−h)2+(𝑦−𝑘)2=𝑟2
𝑟2=41
𝐶𝑒𝑛𝑡𝑒𝑟
𝑟=√41=6.403𝑅𝑎𝑑𝑖𝑢𝑠
𝑥−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 (𝑦=0)
𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 (𝑥=0)
(0,0)(0 ,−8)
(0−5)2+(𝑦+4)2=4125+(𝑦+4)2=41(𝑦+4 )2=16y+4=± 4 𝑦=0 ,−8
(0,0)(10,0)
(𝑥−5)2+(0+4)2=41(𝑥−5)2+16=41(𝑥−5)2=25𝑥−5=±5 𝑦=0 ,10
(0,0)
(0 ,−8)
(10,0)
𝑟=6.403(5 ,−4)
Section 2.4 – Circles
Find the coordinates of the center, the radius, the equation of the circle and the intercepts.
Examples
( 8+22, 4+2
2 )
(𝑥−h)2+(𝑦−𝑘)2=𝑟2
𝑟2=10
𝐶𝑒𝑛𝑡𝑒𝑟
𝑅𝑎𝑑𝑖𝑢𝑠
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑜𝑓 h𝑡 𝑒𝑐𝑖𝑟𝑐𝑙𝑒
(8,4 )(5,3)
(2,2)
(5,3)
𝑟=√(8−5)2+(4−3)2
𝑟=√9+1𝑟=√10=3.162
(𝑥−h)2+(𝑦−𝑘)2=𝑟2
(𝑥−5)2+(𝑦−3)2=10
Section 2.4 – Circles
Find the coordinates of the center, the radius, the equation of the circle and the intercepts.
Examples (𝑥−h)2+(𝑦−𝑘)2=𝑟2
(4,0) (6,0)
(8,4 )(5,3)
(2,2) (𝑥−5)2+(0−3)2=10
𝑥−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 (𝑦=0)
(𝑥−5)2+9=10(𝑥−5)2=1𝑥−5=±1𝑥=4 ,6
(𝑥−5)2+(𝑦−3)2=10
Section 2.4 – Circles
Find the coordinates of the center, the radius, and the intercepts.Examples (𝑥−h)2+(𝑦−𝑘)2=𝑟2
𝐶𝑒𝑛𝑡𝑒𝑟 :
𝑥2+ 𝑦2−4 𝑥+8 𝑦=20𝑥2−4 𝑥+𝑦2+8 𝑦=20
(2 ,−4)
−42 =−2(−2)2=4 8
2=442=16
(𝑥¿¿2−4 𝑥+4)+ (𝑦2+8 𝑦+16 )=20+4+16¿(𝑥−2)2+(𝑦+4 )2=40
𝑅𝑎𝑑𝑖𝑢𝑠 :√ 40=2√10=6.325
Section 2.4 – Circles
Find the coordinates of the center, the radius, and the intercepts.Examples (𝑥−h)2+(𝑦−𝑘)2=𝑟2
𝑥2+ 𝑦2−4 𝑥+8 𝑦=20
(𝑥−2)2+(0+4 )2=40𝑥−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 (𝑦=0)
(𝑥−2)2+16=40(𝑥−2)2=24𝑥−2=±√24
𝑥=2.899 ,6.899
(𝑥−2)2+(𝑦+4 )2=40
𝑥=2±√24
(0−2)2+(𝑦+4)2=40𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 (𝑥=0)
4+(𝑦 +4 )2=40(𝑦+4 )2=36𝑦+4=±6
𝑥=−10 ,2𝑥=−4±6