CONIC SECTIONS 1 Circle 2 Ellipse 3 Parabola

CONIC SECTIONS 1 - Circle 2 - Ellipse 3 - Parabola 4 - Hyperbola Done by : - Mohamed Yousef Al. Qursi - Hamad Ali Saleh - Hassan Taher - Mohamed Almansoori 12 -7

Task 1 / Introduction • The circle is one of the most common shapes in our daily life, and indeed the universe. Planets, the movement of the planets, natural cycles, and natural shapes - there are circles absolutely everywhere. The circle is one of the most complex shapes, and indeed the most difficult for man to create, yet nature manages to do it perfectly. The centers of flowers, eyes, and many more things are circular and we see them in our every day life • An ellipse is a shape that is formed when a cylinder is cut at an angle. If you tilt a glass of water so that the surface is no longer horizontal, the resulting shape of the water is an ellipse. Ellipses can also be seen when a hula hoop or tire of a car is turned askew. Though these are real life examples of optical ellipses, the ellipse also has uses in real life.

Task 1 / Introduction 2 • Parabolas are used in flashlight reflectors, satellite dish antennas, radio telescopes, and in other devices. They are also seen in phenomenon such as trajectory of a object having some mass and under constant gravitational pull. basketball (or any other ball) flying through the air, • A hyperbola is the set of all points such that the difference of the distances between any point on the hyperbola and two fixed points is constant. The two fixed points are called the foci of the hyperbola.

1 Task / Picture Album 1 - Circle 2 - Parabola 3 - Ellipse 4 - Hyperbola

2 Task / Conics Definitions Conics Definition Parabola The set of all points in a plane that are the same distance from a given point called focus and given line called the directrix Circle Is the set of all points in a plane that are equidistant from a given point in the plane, called center Ellipse Is the set of all points in the plane such that the sum of the distance from two fixed points is constant Hyperbola Is the set of all points in a plane such that the absolute value of the differences of the distances from the foci is constant

Task 2 /Standard Forms of Equations of Conic Sections : Type of conics Equations (x – h)2 + (y – k)2 = r 2 Center is (h, k). Circle Radius is r. Parabola with horizontal axis Vertex is (h, k). Focus is (h + p, k). Directrix is the line x = h – p. Axis is the line y = k. Parabola with Vertical axis X= a (y-k) 2 + h Y= a (x-h) 2 + k Vertex is (h, k). Focus is (h, k + p). Directrix is the line y = k – p. Axis is the line x = h.

Task 2 Type of conics Equations Ellipse with horizontal major axis Center is (h, k). Length of major axis is 2 a. Length of minor axis is 2 b. Distance between center and either focus is c with c 2 = a 2– b 2, a > b > 0. Ellipse with Vertical major axis Center is (h, k). Length of major axis is 2 a. Length of minor axis is 2 b. Distance between center and either focus is c with c 2 = a 2– b 2, a > b > 0. Hyperbola with horizontal transverse axis Center is (h, k). Distance between the vertices is 2 a. Distance between the foci is 2 c. c 2 = a 2 + b 2 Hyperbola with vertical transverse axis Center is (h, k). Distance between the vertices is 2 a. Distance between the foci is 2 c. c 2 = a 2 + b 2

Task 3 /Parabola • 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). [hint: use the standard quadratic equation: Y= ax 2+bx+c

Task 3 /Parabola • (0, 3)(-2, 7) and (1, 4) 3 = a(0) + b(0) +c ~> c = 3 • 7=a(-2)2+b(-2)+3 4/2= 4 a/2 -2 b/2 ~> 2 = 2 a – b • 4 = a(1)2+b(1)+3 1 = a+b | a+b=1 2 =2 a - b | 1+b=1 ––––– | b=0 3/3=3 a/3 ~> a = 1 Check : Y=(0)2+3 Y= 3 • Y = ax 2+bx+c Y=(1)x 2 +(0)x+(3) Y= x 2 +3 | (0, 3) |

Task 3 /Parabola The Graph of the function: y=x 2+3

Task 3 /Circle 2+�� 2 =25. • If you have a line equation �� +2�� =2 and circle equation �� How many points the graphs of these two equations have in common. • Now Graphically explore the all cases of line and circle intersections in the plane.

Task 3 /Circle • Line equation : x+2 y=2 • Circle equation : x 2+y 2=25 X+2 y=2 ~ 2 y/2=2 -x/2 • Y= 1 – x/2 Point of inersection( x 2=-4) There will be two points of intersection at x= -4 and x=4. 8 X 2 + ( 1 -x/2)2 = 25 X 2 + 1 – x+ ¼x 2 =25 5/4 x 2 – x + 1 =25 5/4 x 2 -x-24=0 • X 1 = 4. 8 • X 2 = – 4 Point of intersection( x 1=4. 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.

Task 4 / Physics • Height = 20 , a = 80 => 1/-80 Focus = (0, 0) , Center = (0, 20) • Y = 1/-80 (x-0) 2 + 20

Task 4 / Halley's Comet • 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.

Task 4 / Halley's Comet • Hint: the foci are Earth and the Sun with origin in the middle. • Values of 2 c =146/2 c = 73 • a = 30/2 = 15 c 2 = a 2 + b 2 5329 = 225 + b 2 = 5104 • X 2/225 – Y 2/5104= 1