Conic Sections – Circle PDF

SH1712 Conic Sections – Circle Conic Sections Conic sections are curves generated by the intersection of a cone and a plane.  Circle  Parabola  Ellipse  Hyperbola A circle is a conic section obtaine...

SH1712 Conic Sections – Circle Conic Sections Conic sections are curves generated by the intersection of a cone and a plane.  Circle  Parabola  Ellipse  Hyperbola A circle is a conic section obtained by intersecting a cone with a plane perpendicular to its axis of symmetry. It is of a set of points equidistant to a given point called the center. Basic Concepts of a Circle  Center: It is the point wherein all points in a circle are equidistant to.  Circumference: It is the curve that bounds the circle.  Diameter: It is the line crossing the circle and passing through the center. Suppose the 𝑥 +𝑥 𝑦 +𝑦 endpoints of diameter are (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ), then the center is given by ( 1 2 2 , 1 2 2 ).  Radius: It is a straight line drawn from the center to the boundary line or circumference. Its length is half of the length of the diameter.  Chord: It is a straight line joining two (2) points on the circumference points of a circle.  Arc: It is a part of the circumference between two (2) points or a continuous piece of a circle. Standard Equation of a Circle with center at (h, k) and radius 𝒓: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 Graphing a circle STEP 1. Detetermine ℎ, 𝑘, and 𝑟. STEP 2. Locate the center (ℎ, 𝑘). STEP 3. Locate the endpoints of horizontal diameter (ℎ ± 𝑟, 𝑘) and endpoints of vertical diameter (ℎ, 𝑘 ± 𝑟). STEP 4. Draw the circle that passes on these four (4) points and label the graphs (center, radius, and diameter endpoints). Expressing Polynomial Equation of a Circle to Standard Form STEP 1. Transpose the constant term. STEP 2. Group 𝑥 and 𝑦 terms. STEP 3. Complete the square for expressions in 𝑥 and 𝑦. STEP 4. Express the 𝑥 and 𝑦 expressions as perfect squares and simplify the constant terms. 𝑎 2 𝑏 2 𝑎2 +𝑏 2 −4𝑐 𝑥 2 + 𝑦 2 + 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 ⇔ (𝑥 + 2) + (𝑦 + 2) = 4 𝑎 𝑏 √𝑎2 +𝑏2 −4𝑐 Center is at (− 2 , − 2 ) with radius. 2 References: Coburn, J. (2016). Pre-Calculus. 2 Penn Plaza, New York. McGraw Hill Education. Weisstein, E. (n.d.). Conic section. In MathWorld – A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/ConicSection.html 01 Handout 1 *Property of STI Page 1 of 1