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