Math Project Presentation Name Done by: Abdulrahman Ahmed Almansoori Mohammed Essa Suleiman Mohammed Saeed Ahmed Alali

Write an introduction about each conic section, showing what they are used for. 01 Task 1 Parabola:  Parabola: A parabola can be defined as the set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix. The line segment through the focus of a parabola and perpendicular to the axis of symmetry is called the latus rectum. The endpoint of the latus rectum lie on the parabola. Parabola:  Parabola: A parabola can be defined as the set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix. The line segment through the focus of a parabola and perpendicular to the axis of symmetry is called the latus rectum. The endpoint of the latus rectum lie on the parabola. Circle:  Circle: A circle is the set of all points that are equidistant from a given point in the plane, called the center. Any segment with endpoints at the center and a point on the circle is a radius of the circle. Circle:  Circle: A circle is the set of all points that are equidistant from a given point in the plane, called the center. Any segment with endpoints at the center and a point on the circle is a radius of the circle.

01 Task 1 ellipse:  ellipse: An ellipse is the set of all points in a plane such the sum of the distances from two fixed points is constant. These two points are called the foci of the ellipse. Every ellipse has two axes of symmetry, the major axis and minor axis. The axes are perpendicular at the center of ellipse. The foci of an ellipse always lie on the major axis. The endpoints of the major axis are the vertices of the ellipse and the endpoints of the minor axis are the co-vertices of the ellipse. ellipse:  ellipse: An ellipse is the set of all points in a plane such the sum of the distances from two fixed points is constant. These two points are called the foci of the ellipse. Every ellipse has two axes of symmetry, the major axis and minor axis. The axes are perpendicular at the center of ellipse. The foci of an ellipse always lie on the major axis. The endpoints of the major axis are the vertices of the ellipse and the endpoints of the minor axis are the co-vertices of the ellipse. Hyperbola:  Hyperbola: Similar to an ellipse, a hyperbola is the set o all point in plane such that absolute value of the differences of the distances from the foci is constant. Every hyperbola has two axes of symmetry, the transverse axis and conjugate axis. The axes are perpendicular at the center of hyperbola.. The foci of an hyperbola always lie on the transverse axis. The vertices are the endpoints of the transverse axis. The co-vertices are the endpoints of the conjugate axis. Hyperbola:  Hyperbola: Similar to an ellipse, a hyperbola is the set o all point in plane such that absolute value of the differences of the distances from the foci is constant. Every hyperbola has two axes of symmetry, the transverse axis and conjugate axis. The axes are perpendicular at the center of hyperbola.. The foci of an hyperbola always lie on the transverse axis. The vertices are the endpoints of the transverse axis. The co-vertices are the endpoints of the conjugate axis.

Gallery: Go home, library, mall or other public places to find or take pictures for at least a picture for an item that represent each conic section then make a picture album: 01 Task 1 Parabola: Circle

Gallery: Go home, library, mall or other public places to find or take pictures for at least a picture for an item that represent each conic section then make a picture album: 01 Task 1 Ellipse Ellipse Hyperbola Hyperbola

 Comprehensive comparison between conics: Q1) Construct a table that shows the similarities and differences between all types of conics? Comprehensive comparison between conics Showing the following points: Closed or open curve. Its definition Its equation Relation between its center and focus (foci) Other properties Graph it 01 Task 2

Q2)  For circle: 01 Task 2 Answers Shape Closed or open curve Closed curve Definition The set of all points in a plane that are a given distance from a given point, the center. Equation (x-a) 2 + (y-b) 2 =r 2 Relation between it’s center and focus The center and the focus is in the same place Other properties Cutting a circular cone with a plane perpendicular to the symmetry axis of the cone forms a circle. This intersection is a closed curve, and the intersection is parallel to the plane generating the circle of the cone. A circle is also the set of all points that are equally distant from the center. Graph

 For Ellipse : 01 Task 2 Shape Closed or open curve Closed curve Definition A plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve Equation OR Relation between it’s center and focus where (h,k) is the center of the ellipse, rx is the distance from the center of the circle in the x direction and ry is the distance from the center in the y direction. Other properties From the center of the ellipse on the major axis. The major axis is the line of the ellipse that has the biggest distance from the center of the circle. If the major axis is horizontal, 2rx is the length and c2=rx2-ry2. If the major axis is vertical, 2ry is the length and c2=ry2-rx2. Graph

 For Parabola : 01 Task 2 Shape Closed or open curve Open curve Definition A conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface Equation y = ax 2 + bx + c OR x = ay 2 + by + c Relation between it’s center and focus Other properties The standard equation depends on the axis of symmetry. A vertical axis has a focus at (h,k+p) and the equation (x- h)2=4p(y-k). A horizontal axis has a focus at (h+p,k) and the equation (y-k)2=4p(x-h). The vertex is always halfway in between the focus and directrix at a distance p from both. Graph

 For Hyperbola : 01 Task 2 Shape Closed or open curve Open curve Definition A type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set. Equation Or Relation between it’s center and focus The foci in an hyperbola are further from the hyperbola's center than are its vertices Other properties Where (h,k) is the center between the curves and it's two asymptotes go through the points (+a,-b) and (-a,+b) as well as (a,b) and (-a,-b) starting at the center point. Graph

1. 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: = ]. 01 Task 3 First we get C. 3=0+0+C C=3 Then we get the two other equations by replacing C: 7 = a (-2) 2 + b (-2) = 4a – 2b 2 = 2a - b 4 = a (1) 2 + b(1) = a + b 2 = 2a - b 1 = a + b 3 = 3a a = 1, b = 0 First we get C. 3=0+0+C C=3 Then we get the two other equations by replacing C: 7 = a (-2) 2 + b (-2) = 4a – 2b 2 = 2a - b 4 = a (1) 2 + b(1) = a + b 2 = 2a - b 1 = a + b 3 = 3a a = 1, b = 0 4/2 = (4a-2b)/2 2 = 2a – b 1= a + b 3 = 3a a = 1, b = 0 So the equation is going to be: y= x /2 = (4a-2b)/2 2 = 2a – b 1= a + b 3 = 3a a = 1, b = 0 So the equation is going to be: y= x 2 +3

2. Circle: If you have a line equation + 2 = 2 and circle equation = 25. How many points the graphs of these two equations have in common 01 Task 3 x + 2y = 2 2y = 2 – x y = 1 – 0.5 x = 25 x 2 + (1 – 0.5 x) 2 = 25 x 2 + (1 – x x 2 ) = x 2 – x = x 2 – x – 24 = x 2 – x – 24 = 0 (x+4.8) (x+4) = 0 x = -4.8, or x = -4 x + 2y = 2 2y = 2 – x y = 1 – 0.5 x = 25 x 2 + (1 – 0.5 x) 2 = 25 x 2 + (1 – x x 2 ) = x 2 – x = x 2 – x – 24 = x 2 – x – 24 = 0 (x+4.8) (x+4) = 0 x = -4.8, or x = -4 To get Y 1 and Y 2 we will replace X 1 and X 2 in the following equation y = 1 – 0.5 x, so Y 1 =- 1.4 Y 2 =3 So the intersection points are (4.8,-1.4) and (-4,3) To get Y 1 and Y 2 we will replace X 1 and X 2 in the following equation y = 1 – 0.5 x, so Y 1 =- 1.4 Y 2 =3 So the intersection points are (4.8,-1.4) and (-4,3)

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. 01 Task 4 Vertices (0,20) and the Focus at (0,0) It is vertical, k is different in the vertex and focus. So the equation of the parabola is y=a(x-h) 2 +k y=a(x-h) 2 +k Latus rictum = I 1/a I I 1/a I = 80 a = -1/80, because it is open to down Latus rictum = I 1/a I I 1/a I = 80 a = -1/80, because it is open to down y = a(x-h) 2 + k y = -180(x-0) y = -1/80x y = a(x-h) 2 + k y = -180(x-0) y = -1/80x

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. 01 Task 4 c 2 = a 2 + b 2 b 2 = c 2 - a 2 b 2 = (73) 2 - (15) 2 b 2 = 5104 c 2 = a 2 + b 2 b 2 = c 2 - a 2 b 2 = (73) 2 - (15) 2 b 2 = 5104 distance between the sun and the earth = 146 millions Km Centre is (0,0) c = 146/2 = 73 The difference of the distances from the comet to each body is 30. a = 302 = 15 million miles

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