The Parabola. The Parabola in Terms of a Locus of Points

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1 The Parabola Appolonius of Perga (5 B.C.) discovered that b intersecting a right circular cone with a plane slanted the same as the side of the cone, (formall, when it is parallel to the slant height), the intersection provides a resulting curve (conic section) in the shape of a parabola. We have seen that the parabola is the shape of the graph of a quadratic function. Parabolas occur in man applications such as the shape of some mirrors on telescopes, satellite dishes, natural path of a ball thrown The Parabola in Terms of a Locus of Points A parabola is the set of points equidistant from a fied line (the directri) and a fied point (the focus).

2 Parabolas ehibit unusual and useful reflective properties. If a light is placed at the focus of a parabolic mirror (a curved surface formed b rotating a parabola about its ais), the light will be reflected in ras parallel to said ais. In this wa a straight beam of light is formed. It is for this reason that parabolic surfaces are used for headlamp reflectors. The bulb is placed at the focus for the high beam and a little above the focus for the low beam. To arrive at a general equation of a parabola in standard position (verte is at the origin) and the graph opens up we will use the locus definition to assist us. First let the focus be on the -ais at F(0,p), then since the parabola is the locus of points equidistant from a line called the directri, the equation of the directri must be the line p. P Let P, D(,-p) be a point on the parabola, then 0 PF PD p p p p 4p More familiar form

3 The General Equation of a Parabola with -Ais as the Ais of Smmetr The general equation of a parabola with the -ais as the ais of smmetr and focus at (0, p) and directri at =-p is or 4p. The General Equation of a Parabola with -Ais as the Ais of Smmetr The general equation of a parabola with the -ais as the ais of smmetr and focus at (p, 0) and directri at =-p is p 4p. Note: The advantage of write the parabola as a b. format is the when given the verte at (a, b), we can Note: The parabola opens up or to the right if p>0, down or to the left if p<0. The distance p, from verte to the focus, is called the focal length. Note: If the verte is not at the origin, we can appl the appropriate translations to arrive at the equation. Eample Equivalent to 8 State the directri and the focus of the parabola 8 The equation is written in the form If ou like ou can use 4 p Therefore 4p=8, so p=. Therefore the focus is (0,) and the directri is =-

4 Eample The general form of a parabola is and graph it. 5. Write the standard form of this conic Change into standard form b dividing b rearranging This is in form 4p, which is a parabola that opens to the right The parabola opens to the right, and its verte is (-3,-).

5 Eample Find the equation for the locus of points that are equidistant from the point (0,), and from the line =-4. State the verte and the focal length of the curve. Let P(,) be a point on the curve and D(,-4) be a point on the directri. The distance from the focus to P is the same as the distance from the D to P. That is PF=PD. PF PD The verte is (0,-), and the focal length is 3 (=43)

6 Eample Determine the directri, the focus, and the equation of the parabola that passes through the point, 0, has verte (0, 0), and has verte on the -ais. Sketch the graph of the parabola, and label its focus and directri. Since the focus is one the -ais, the equation has the form 4, 0 is on the graph, we have Therefore the focus is parabola is 0. p 0 p 0 4p p 5 p. Now, since 5,0 and the directri is the line 5, and the equation of the

7 Eample A radio dish at the VLA ( Ver Large Arra) at Socorro, New Meico has the shape of a parabolic dish (the cross section through the centre of the dish is a parabola). This dish is about feet deep at the centre and has a diameter of 8 feet. How far from the verte of the parabolic dish should the receiver be placed so that it is in the focus? We will draw a cross section of the dish with the parabola s verte on the -ais. This gives us an equation of the form 4. Because the dish is 8 feet across, and feet deep, the point (4, ) is on the parabola. Therefore 4 p 688p p 35 Thus the focus is (0, 35), or the receiver should be placed 35 feet from the verte (bottom of the dish).