Circles
Conic Sections
Circles
• Algebraic definition: set of all points the same distance from a point.
• Center of the circle: the point in the center of the circle….
• Radius: distance from the center to the edge of the circle
• 𝑥−ℎ
2
+ 𝑦−𝑘
2
= 𝑟 2 , (ℎ, 𝑘) is the center of the circle with radius 𝑟.
• Standard form for circles
Practice with Circles
• Describe the circle 𝑥 − 3
2
+ 𝑦+2
2
= 25.
• Circle with center (3, −2) and a radius of 5.
• Give the equation of a circle whose center is (−1, 7) and whose
radius is 3.
• 𝑥+1
2
+ 𝑦−7
2
=9
Practice with Circles
• Describe the circle 𝑥 2 + 𝑦 2 + 6𝑥 − 4𝑦 − 5 = 0.
• How can we get this equation to match our
standard form?
• Think back to how we have solved quadratics
previously…
• 𝑥 2 + 6𝑥 + 𝑦 2 − 4𝑦 = 5
• 𝑥 2 + 6𝑥 + 9 + 𝑦 2 − 4𝑦 + 4 = 5 + 9 + 4
• 𝑥+3
2
+ 𝑦−2
2
= 18
• Circle has a center of (−3, 2) and radius 3 2
Practice with Circles
• Describe the circle 𝑥 − 3
2
+ 𝑦+2
2
= 25.
• Circle with center (3, −2) and a radius of 5.
• Give the equation of a circle whose center is (−1, 7) and whose
radius is 3.
• 𝑥+1
2
+ 𝑦−7
2
=9
Practice with Circles
• Give the equation of a circle
whose diameters has endpoints
of (−4, 1) and (8, −7).
• What is the definition of
“diameter”?
• i.e. what is in the middle of the
diameter?
• Midpoint:
−4+8 1+ −7
,
2
2
= (2, −3)
• What is the length of the radius?
• Distance:
𝑥2 − 𝑥1
2
+ 𝑦2 − 𝑦1
2
• Find the distance between the
center and a point on the circle.
•
2 − −4
2
+ −3 − 1
2
• Now write the equation:
• 𝑥−2
2
+ 𝑦+3
2
= 52
= 52
Half Circles
• Uses the square root of our standard
form to get the top (positive) or
bottom (negative) half of a circle.
• Imagine the graph of 𝑥 2 + 𝑦 2 = 9
• Solve for y: 𝑦 = 9 − 𝑥 2
• Or: 𝑦 = − 9 − 𝑥 2