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Conics

Conics. Conic Sections. The Ellipse. Tycho Brahe Planetarium in Copenhagen. Planetary Orbit. Atoms. Reflection of light and sound waves.

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Conics

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  1. Conics

  2. Conic Sections

  3. The Ellipse Tycho Brahe Planetarium in Copenhagen

  4. Planetary Orbit

  5. Atoms

  6. Reflection of light and sound waves This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

  7. Whispering galleries St. Paul’s Cathedral, London

  8. The Parabola Galileo and cannoneers

  9. Reflective properties Focus

  10. Reflecting Telescopes Subaru Telescope

  11. The Hyperbola

  12. Lamp Shade

  13. Cooling Tower of a steam power plant Nuclear Power Plant in Indiana, USA

  14. Loran (Long Range Navigation). 

  15. Gravitational Slingshot

  16. General Equation of a Circle • What is the equation of a circle • What is the centre and radius? • If we expand this, we get the general equation • Going backwards and completing the square we get the general equation of a circle again. (x - a)2 + (y – b)2 = r2 centre (a,b) radius r x2 + y2 + 2ax + 2by + c = 0

  17. Example 1. Complete Square to find centre and radius • For the circle x2 - 4x + y2 + 6y - 4 = 0, find the coordinates of the centre and the radius x2 –4x + y2 + 6y – 4 = 0 HALF THE X TERM HALF THE Y TERM ( x – 2 )2 – ( -2 )2 + ( y + 3 )2 – ( 3 )2– 4 = 0 ( x – 2 )2 – 4 + ( y + 3 )2 – 9 – 4 = 0 ( x – 2 )2 + ( y + 3 )2 = 17 Center? Radius? (2,-3) √ 17

  18. General Equation: x2 + y2 + 2ax + 2by + c = 0 Example 2. Write the general equation of the circle with Centre (4, -3) passing through (1, 1) (x - 4)2 + (y + 3)2 = r2 Substituting (4, -3) (1 - 4)2 + (1 + 3)2 = r2 When x=1 and y=1 25 = r2 Page 358 # 1 - 9 5 = r (x-4)2 + (y+3)2 = 25 Substituting r2 =25 x - 8x + 16 + y + 6x + 9 = 25 Expand x - 8x + y + 6x = 0 Expand to find General Soln.

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