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Conic Sections

Conic Sections. Grade 12 mathematics project Term 2. Dony by : Mohammed Rashed Altaboor Rashed Yousef Lehmaidi Hamad J asem A lmansoori Yousef A bdulla A lnuaimi G: 12 – Section: 6. I ntroduction.

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Conic Sections

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  1. Conic Sections Grade 12 mathematics project Term 2 • Dony by: • Mohammed RashedAltaboor • RashedYousefLehmaidi • HamadJasemAlmansoori • YousefAbdulla Alnuaimi • G: 12 – Section: 6

  2. Introduction • In mathematics, a conic section or is a curve obtained as the intersection of a cone with a plane. • The four type of conic section is parabola, circle, ellipse and Hyperbola. Parabolas are used in car headlight, mirror in reflecting telescopes and television and radio antennae. Circles are used in car wheels, clocks, bottle covers and headphones. Ellipses are used in race tracks, spotlight on a planar surface, tanks and the orbit of the planets is ellipse. Hyperbola are used in Lampshade, Gear transmission, Sonic Boom and Dulles Airport. There is many other uses of conic section which makes them very important in our daily life.

  3. History • In 300 B.C. Menaechmus, and Euclid studied conic sections just for the beauty of the mathematics.  In 200 B.C. Apollonius first used the terms parabola, ellipse and hyperbola.  For many years the usefulness of these slices of the cone was unknown.  Now, here you are in the year 2004 A.D. studying these same quadratic equations.  How do conic sections help us describe and predict natural phenomena?  How are we able to apply the properties of these ancient curves in this technological age?

  4. All conic sections has intersection of a plane and a cone. The circle is a conic section where the plane is perpendicular to the axis of the cone. The special case of a point is where the vertex of the cone lies on the plane it has infinite number of asymptotes. The ellipse is a conic section where the plane is not perpendicular to the axis, but its angle is less than one of the napes. The special case of a point is where the vertex of the cone lies on the plane it has 2 asymptote. The parabola is a conic section where the plane is parallel to one of the napes. The special case of two intersecting lines is where the vertex of the cone lies on the plane it has 1 asymptote. The hyperbole is a conic section where the angle of the plane is greater than on of the napes. There are two sides to the hyperbole. The special case of two lines intersecting is where the vertex of the cone lies on the plane it has 2 asymptote. comparison between conics

  5. Task 3 (Parabola) -Addition of the equations 2a – b + a + b = 2 + 1 3a = 3 a = 3 a + b = 1 1 + b = 1 b = 0 Final Equation: 𝑦 = ax2 + b𝑥 + c 𝑦 = 1𝑥2 + 0(𝑥) + 3 𝑦 = x2 + 3 You can find the equation of a line by knowing two points from that line, know to find and equation of parabola you need to know three points. Find the equation of a parabola that pass through (0,3), (-2, 7) and (1, 4). 𝑦 = a𝑥2 + b𝑥 + c Since it passes through (0,3) 3= a(0)2 + b(0) + c c = 3 𝑦 = a𝑥2 + b𝑥 + 3 Since it passes through (-2,7) 7 = a(-2)2 + b(-2) + 3 7 = 4a – 2b + 3 4 = 4a – 2b ( 4a – 2b = 4 )/2 2a – b = 2 Passes Through (1,4) 4 = a(1)2 + b(1) + 3 1 = a + b

  6. Task 3 (Circle) If you have a line equation X + 2y = 2 and circle equation X2 + Y2 = 25. How many points the graphs of these two equations have in common. 𝑥 + 2𝑦 = 2 𝑥 = 2 - 2𝑦 𝑥2 + 𝑦2 = 25 (2 - 2𝑦)2 + 𝑦2 = 25 (2 - 2𝑦) (2 - 2𝑦) + 𝑦2 = 25 4 - 4𝑦 - 4𝑦 + 4𝑦2 + 𝑦2= 25 5𝑦2 - 8𝑦 - 21 = 0 𝑦 = 3 𝑦 = -1.4 𝑥 = 2 – 2(3) or 𝑥 = 2 – 2(-1.4) 𝑥 = 4 𝑥 = 4.8 Two Points in common: (-4,3) (4.8,-1.4)

  7. Task 3 (Circle) cont. Now Graphically explore the all cases of line and circle intersections in the plane. Circle: 𝑥2 + 𝑦2 = 25 - Center(0,0), Radius: 5 Line: 𝑥 + 2𝑦 = 2

  8. Task 4 (Physics) The path of any thrown ball is parabola. Suppose a ball is thrown from ground level, reach a maximum height of 20 meters of and hits the ground 80 meters from where it was thrown. Find the equation of the parabolic path of the ball, assume the focus is on the ground level. Equation of the parabolic path: 𝑦 = a(𝑥 - h) + k, a<0 Since focus is at origin h=0 Vertex (h,k) = (0,20) Since the parabola opening downward a < 0

  9. Task 4 ( Halleys’s Comet) The distance between the sun and earth is 146 million km. Equation of hyperbola: It takes about 76 years to orbit the Sun, and since it’s path is an ellipse so we can say that its movements is periodic. But many other comets travel in paths that resemble hyperbolas and we see it only once. Now if a comet follows a path that is one branch of a hyperbola. Suppose the comet is 30 million miles farther from the Sun than from the Earth. Determine the equation of the hyperbola centered at the origin for the path of the comet. Hint: the foci are Earth and the Sun with origin in the middle.

  10. References • Book: Algebra 2 SE: ( pages:599 to 631) • http://wiki.answers.com/Q/What_are_the_differences_between_four_conic_sections

  11. Thank You For Watching Our Presentation

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