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Conic Sections The Parabola Introduction • Consider a cone being intersected with a plane Note the different shaped curves that result Introduction They can be described or defined as a set of points which satisfy certain conditions • We will consider various conic sections and how they are described analytically Parabolas Hyperbolas Ellipses Circles Parabola • Definition A set of points on the plane that are equidistant from A fixed line (the directrix) and A fixed point (the focus) not on the directrix Parabola • Note the line through the focus, perpendicular to the directrix Axis of symmetry • Note the point midway between the directrix and the focus Vertex View Geogebra Demonstration Equation of Parabola • Let the vertex be at (0, 0) Axis of symmetry be y-axis Directrix be the line y = -p (where p > 0) Focus is then at (0, p) • For any point (x, y) on the parabola Distance = x 0 y p 2 2 ( x, y ) Distance = y + p Equation of Parabola • Setting the two distances equal to each other x 0 y p 2 2 y p . . . simplifying . . . x2 4 p y • What happens if p < 0? 2 • What happens if we have y 4 p x ? Working with the Formula • Given the equation of a parabola y = ½ x2 • Determine The directrix The focus • Given the focus at (-3,0) and the fact that the vertex is at the origin • Determine the equation When the Vertex Is (h, k) • Standard form of equation for vertical axis of 2 symmetry x h 4 p y k • Consider What are the coordinates of the focus? What is the equation of the directrix? (h, k) When the Vertex Is (h, k) • Standard form of equation for horizontal axis 2 of symmetry y k 4 p x h • Consider What are the coordinates of the focus? What is the equation of the directrix? (h, k) Try It Out • Given the equations below, What is the focus? What is the directrix? 4 x2 12 x 12 y 7 0 ( x 3) ( y 2) 2 x y 4y 9 0 2 Another Concept • Given the directrix at x = -1 and focus at (3,2) • Determine the standard form of the parabola Applications • Reflections of light rays Parallel rays strike surface of parabola Reflected back to the focus View Animated Demo Spreadsheet Demo How to Find the Focus Proof of the Reflection Property Build a working parabolic cooker MIT & Myth Busters Applications • Light rays leaving the focus reflect out in parallel rays Used for Searchlights Military Searchlights Assignment • See Handout • Part A 1 – 33 odd • Part B 35 – 43 all