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math conic sections.pptx

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COORDINATE GEOMETRY II
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math conic sections.pptx

math conic sections eloboration on parabola

math conic sections eloboration on parabola

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math conic sections.pptx

1. 1. BY S.VARSHA CLASS XI SRI JAYENDRA SWAMIGAL VIDHYA KENDRA V.M CHATRAM
2. 2.  Conic sections, also called conic, in geometry, any curve produced by the intersection of a plane and a right circular cone. .  Conics may also be described as plane curves that are the paths (loci) of a point moving so that the ratio of its distance from a fixed point (the focus) to the distance from a fixed line (the directrix) is a constant, called the eccentricity of the curve.  If the eccentricity is zero, the curve is a circle; if equal to one, a parabola; if less than one, an ellipse; and if greater than one, a hyperbola  Every conic section corresponds to the graph of a second degree polynomial equation of the form Ax2 + By2 + 2Cxy + 2Dx + 2Ey + F = 0
3. 3.  Parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. THE VERTEX The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved THE FOCAL LENGTH The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length. THE LATUS RECTUM The latus rectum is the chord of the parabola that is parallel to the directrix and passes through the focus. THE DIRECTRIX The directrix of a parabola is a line that is perpendicular to the axis of the parabola. The directrix of the parabola helps in defining the parabola.
4. 4.  The equation of parabola can be expressed in two different ways, such as the standard form and the vertex form. The standard form of parabola equation is expressed as follows:  f(x) = y= ax2 + bx + c GRAPHING THE PARABOLA  Two points define a line. Since parabola is a curve-shaped structure, we have to find more than two points here, to plot it.  We need to determine at least five points as a medium to design a leasing shape.  In the beginning, we draw a parabola by plotting the points.  Suppose we have a quadratic equation of the form y=ax2+ bx + c, where x is the independent variable and y is the dependent variable.  We have to choose some values for x and then find the corresponding y-values. Now, these values of x and y values will provide us with the points in the x-y plane to plot the required parabola.  With the help of these points, we can sketch the graph. EXAMPLE GRAPHS
5. 5. PROBLEM  An engineer designs a satellite dish with a parabolic cross section. The dish is 5 m wide at the opening, and the focus is placed 1 2 . m from the vertex  (a) Position a coordinate system with the origin at the vertex and the x -axis on the parabola’s axis of symmetry and find an equation of the parabola.  (b) Find the depth of the satellite dish at the vertex.  SOLUTION FOR (A): The equation for the given parabola is y2 = 4ax
6. 6.  SOLUTION FOR (B)  y2 = 4ax  here a = 1.2  y2 = 4(1.2)x  y2 = 4.8 x  The parabola is passing through the point (x, 2.5)  (2.5)2 = 4.8 x  x = 6.25/4.8  x = 1.3 m  Hence the depth of the satellite dish is 1.3 m.
7. 7. There are many applications of parabola in real life,  Satellite dishes use parabolas to help reflect signals that are subsequently sent to a receiver. Because of the reflecting qualities of parabolas, signals sent directly to the satellite will bounce off and return to the receiver after bouncing off the focus .  The reflecting properties of parabolas are used in several heaters. The heat source lies in the centre, with parallel beams concentrating the heat.  parabolic arches have the most thrust at their bases and can span the greatest distances when the weight is spread uniformly over the arch.Parabolas and similar curves are often used to make pleasing arches and shapes in buildings and bridges.