Conic Sections Maths Project Work Made by:- GUNJAN XI ‘A’: Conic Sections Maths Project Work Made by:- GUNJAN XI ‘A’
Conic Sections - Introduction: Conic Sections - Introduction A conic is a shape generated by intersecting two lines at a point and rotating one line around the other while keeping the angle between the lines constant.
Conic Sections - Introduction: Conic Sections - Introduction The resulting collection of points is called a right circular cone. The two parts of the cone intersecting at the vertex are called nappes. Vertex Nappe
Conic Sections - Introduction: Conic Sections - Introduction A “conic” or conic section is the intersection of a plane with the cone. The plane can intersect the cone at the vertex resulting in a point .
What are conic sections?: What are conic sections? Conic sections are lines that define where a flat plane intersects with a double cone, which consists of two cones that meet at one another’s tip.
Conic Sections and Degenerate Conic Sections: Conic Sections and Degenerate Conic Sections
Conic Sections and Degenerate Conic Sections (cont’d): Conic Sections and Degenerate Conic Sections (cont’d) Animation
Conic Sections: Conic Sections CIRCLE
Slide 9: The plane can intersect the cone perpendicular to the axis resulting in a circle .
Slide 10: Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone. A circle is formed when i.e. when the plane is perpendicular to the axis of the cones.
Slide 11: Circles are the easiest to figure out and graph out of the four conic sections. The formula for the radius of a circle is x 2 + y 2 = r 2 , with (0,0) as the center point of the circle. The Standard Form of a circle with a center at (h,k) and a radius r, is…….. Example :- center (3,3) radius = 2
Conic Sections: Conic Sections ELLIPSE
Conic Sections - ELLIPSE: Conic Sections - ELLIPSE The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse .
Slide 14: An ellipse is the set of all points in a plane whose distance from two fixed points in the plane have a constant sum. The fixed points are the foci of the ellipse. The line through the foci is the focal axis . The point on the focal axis midway between the foci is the center . The points where the ellipse intersects its axis are the vertices of the ellipse. P’(x,y) P’’(x,y)
Slide 15: Let PF 1 +PF 2 = 2 a where a > 0 standard equation of an ellipse
Slide 16: vertex major axis = 2a minor axis = 2b lactus rectum length of semi-major axis = a length of the semi-minor axis = b length of lactus rectum =
Slide 17: Other form of Ellipse
Eccentricity of an Ellipse: Eccentricity of an Ellipse
Elliptical Orbits Around the Sun: Elliptical Orbits Around the Sun
Conic Sections: Conic Sections PARABOLA
Conic Sections - Parabola: Conic Sections - Parabola The plane can intersect one nappe of the cone at an angle to the axis resulting in a parabola .
Slide 23: The intersection of a plane with one nappe of the cone is a parabola. A parabola is the set of all points in a plane equidistant from a particular line (the directrix ) and a particular point (the focus ) in the plane.
Slide 24: From the definition of parabola, PF = PN standard equation of a parabola focus F( a ,0) P( x , y ) M(- a ,0) x y O
Slide 25: mid-point of FM = the origin (O) = vertex length of the latus rectum = LL’= 4a vertex latus rectum (LL’) axis of symmetry
Slide 26: Other forms of Parabola
Slide 27: Other forms of Parabola
Graphs of x2=4py: Graphs of x 2 =4 py
Parabolas with Vertex (0,0): Parabolas with Vertex (0,0) Standard equation x 2 = 4 py y 2 = 4 px Opens Upward or To the right downward or to the left Focus (0, p ) ( p ,0) Directrix y = - p x = - p Axis y -axis x -axis Focal length p p Focal width |4 p | |4 p |
Graphs of y2 = 4px: Graphs of y 2 = 4 px
Paraboloid Revolution: Paraboloid Revolution Parabola
Paraboloid Revolution: Paraboloid Revolution A paraboloid revolution results from rotating a parabola around its axis of symmetry as shown at the right.
Paraboloid Revolution: Paraboloid Revolution They are commonly used today in satellite technology as well as lighting in motor vehicle headlights and flashlights.
Paraboloid Revolution: Paraboloid Revolution The focus becomes an important point. As waves approach a properly positioned parabolic reflector, they reflect back toward the focus. Since the distance traveled by all of the waves is the same, the wave is concentrated at the focus where the receiver is positioned.
Conic Sections: Conic Sections Hyperbola
Conic Sections -Hyperbola: Conic Sections -Hyperbola The plane can intersect two nappes of the cone resulting in a hyperbola .
Hyperbola - Definition: Hyperbola - Definition A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal axis . The point on the focal axis midway between the foci is the center . The points where the hyperbola intersects its focal axis are the vertices of the hyperbola. | d 1 – d 2 | is a constant value. If the length of d 2 is subtracted from the left side of d1 , what is the length which remains? | d 1 – d 2 | = 2a
Slide 39: P’(x,y) Let |PF 1 -PF 2 | = 2 a where a > 0
Slide 40: standard equation of a hyperbola
Slide 41: vertex transverse axis conjugate axis lactus rectum length of lactus rectum = length of the semi-transverse axis = a length of the semi-conjugate axis = b
Slide 42: Other form of Hyperbola :
Eccentricity of a Hyperbola: Eccentricity of a Hyperbola
Hyperbola: Hyperbola The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.
Where are the Hyperbolas?: Where are the Hyperbolas? A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.
How are conic sections used in the real world?: How are conic sections used in the real world? Believe it or not, conic sections really can be used in real-world situations. The circle formula can be used to figure out how long it takes for the blast from a supernova to reach out to certain distances in space. The ellipse formula can be used to find out the length and width of a running track. The hyperbola formula can be used to figure out the angles of light coming from a lighthouse. Parabolas can be used to measure things like suspension bridges.