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Introduction to Conic Sections & The Circle: Study Circles in Detail

Learn about conic sections formed by plane intersection with cones. Explore circles defined by points equidistant from a center. Solve circle problems and equations with key formulas and concepts.

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Introduction to Conic Sections & The Circle: Study Circles in Detail

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  1. 11.1 Intro to Conic Sections & The Circle

  2. What is a “Conic Section”? A curve formed by the intersection of a plane and a double right circular cone. We will study these in detail one at a time!

  3. Circles : set of all points in a plane at a fixed distance from a fixed point (center) (radius) P(x, y) r Center C(h, k) Any point on circle P(x, y) By distance formula: C(h, k) standard form of a circle Check out the problems around the room. Work together and answer them all!

  4. Find center & radius. x2 + y2 + 8x – 10y = 23 C(–4, 5) r = 8 Determine an equation of a circle congruent to the graph of x2 + y2 = 16 and translated 3 units right and 1 unit down. (x – 3)2 + (y + 1)2 = 16 The general form of a circle is x2 + y2 + Dx+ Ey + F = 0. *In completing the square if r > 0  circle r = 0  degenerate circle / point circle r < 1  the empty set (not possible) Determine what 3x2 + 3y2 – 30x + 18y + 178 = 0 represents. empty set

  5. Determine the equation of the circle that passes through these three points: (5, 3), (–1, 9), (3, –3). • *Use x2 + y2 + Dx + Ey +F = 0 • here’s a hint … for (5, 3): 25 + 9 + 5D + 3E + F = 0 • x2+ y2 + 4x – 4y – 42 = 0  (x + 2)2 + (y – 2)2 = 50 • Determine an equation of a circle that satisfies the center at (2, 3) tangent to line 5x + 6y = 14. • *remember! Distance from a point to a line (x1, y1) d Ax + By + C = 0

  6. Homework #1101 Pg538 #5, 7, 15, 21, 22, 24–26, 30–32, 34, 36, 38, 41, 45, 47, 49, 51

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