Parabola

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Parabola

  1. 1. Parabola Analytical Geometry T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
  2. 2. Parabola  A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane.  The fixed line is called the directrix of the parabola and the fixed point is called the focus. ‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when you throw a ball in the air.  A line through the focus and perpendicular to the directrix is called the axis of the parabola.  The point of intersection of parabola with the axis is called the vertex of the parabola. Directrix Focus Vertex © iTutor. 2000-2013. All Rights Reserved
  3. 3. Standard equations of parabola  The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis or y-axis. The four possible such orientations of parabola are shown below in Fig. (a) to (d). F (a, 0)O Y’ XX’ Y x=-a y2 =4ax F (-a, 0) O Y’ XX’ Y x=a y2 = - 4ax(a) (b) © iTutor. 2000-2013. All Rights Reserved If the coefficient of x is positive If the coefficient of x is negative
  4. 4. We will derive the equation for the parabola shown above in Fig (a) with focus at (a, 0) a> 0; and directrix x= –a as below: F (0, a) O Y’ XX’ Y y = -a x2 =4ay(c) F (0, -a) O Y’ XX’ Y y = a x2 = - 4ay(d) © iTutor. 2000-2013. All Rights Reserved If the coefficient of y is positive If the coefficient of y is negative
  5. 5. Let F be the focus and l the directrix, FM  directrix and bisect FM at the point O. Produce MO to X. By the definition of parabola, the mid-point O is on the parabola and is called the vertex of the parabola. Take O as origin, OX the x-axis and OY  to it as the y-axis. Let the distance from the directrix to the focus be 2a. Then, the coordinates of the focus are (a, 0), and the equation of the directrix is x +a= 0 as in Figure. Let P(x, y) be any point on the parabola such that PF = PB, …………….... (1) where PB  l. The coordinates of B are (–a, y). F (a, 0)O Y’ X Y x=-a M l B(-a , y) P (x , y) © iTutor. 2000-2013. All Rights Reserved
  6. 6. By the distance formula, we have and Since PF = PB, we have Hence, any point on the parabola satisfies Conversely, let P(x, y) satisfy the equation (2) 22 )( yaxPF  2 )( axPB  222 )()( axyax  222 )()( axyax i.e., 22222 22 aaxxyaaxx  axy 4 2  or or --------------- (2)axy 4 2  axaxyaxPF 4)()( 222  PBax  2 )( --------------- (3) © iTutor. 2000-2013. All Rights Reserved
  7. 7. and so P(x, y) lies on the parabola. Thus, from (2) and (3) we have proved that the equation to the parabola with vertex at the origin, focus at (a,0) and directrix x= –a is Similarly, we can derive the equations of the parabolas in: Fig (b) as y2 = – 4ax, Fig (c) as x2 = 4ay, Fig (d) as x2 = – 4ay, These four equations are known as standard equations of parabolas axy 4 2  © iTutor. 2000-2013. All Rights Reserved
  8. 8. From the standard equations of the parabolas, we have the following observations: 1. Parabola is symmetric with respect to the axis of the parabola. If the equation has a y2 term, then the axis of symmetry is along the x-axis and if the equation has an x2 term, then the axis of symmetry is along the y-axis. 2. When the axis of symmetry is along the x-axis the parabola opens to the : a. Right if the coefficient of x is positive, b. Left if the coefficient of x is negative. 3. When the axis of symmetry is along the y-axis the parabola opens: a. Upwards if the coefficient of y is positive. b. Downwards if the coefficient of y is negative. © iTutor. 2000-2013. All Rights Reserved
  9. 9. Latus Rectum  Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola . To find the Length of the latus rectum of the parabola y2= 4ax By the definition of the parabola, AF = AC. But AC = FM = 2a Hence AF = 2a. And since the parabola is symmetric with respect to x-axis AF = FB and So AB = Length of the latus rectum = 4a. O X Y Focus Latus rectum F (a, 0)O Y’ X Y M B A C © iTutor. 2000-2013. All Rights Reserved
  10. 10. The End Call us for more Information: www.iTutor.com Visit 1-855-694-8886

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