6.1 Introduction The General Quadratic Equation in x and y has the form: Ax 2 Bxy Cy 2 Dx Ey F 0 Where A, B, C, D, E, F are constants. The graphs of these equations are called Conic Sections, or simply Conics. There are three basic distinct figures that result from the intersection of a plane with a double-napped cone: 1.) Parabola 2.) Ellipse 3.) Hyperbola 6.2 Parabolas A parabola is the set of points in a plane that are equidistant from a given point, called the focal point, and a given line that does not contain the focal point called the directrix. The axis and vertex of a parabola: The axis of the parabola is the line that goes through the focal point and is perpendicular to the directrix. The point of intersection of the axis and the parabola is the vertex. A standard form for the equation of the parabola is derived by superimposing an xy-coordinate system so that the axis of the parabola corresponds with the y-axis, and the vertex of the parabola is located midway between the focal point and the directrix (at the origin). Focal Point vertex Directrix Axis figure not drawn to scale If the focal point is located at (0, c), then the directrix has equation y = -c Note: By the definition of the parabola, the distance between any point (x, y) on the parabola and the focal point is equal to the distance between that point (x, y) and the directrix. Standard Position Parabolas: A parabola with focal point at (0, c), vertex at (0, 0), and directrix y = -c is said to be in standard position with axis along the y-axis and has equation: 1 2 y x 4c Similarly, a parabola with focal point at (c, 0), vertex at (0, 0), and directrix x = -c is in standard position with axis along the x-axis and has equation: 1 2 x y 4c Example 1: Find an equation of the parabola with focal point (0, -2) and directrix y = 2. Visualize or draw the given information. Since we see that the focal point has c = -2 and directrix y = 2, the value of the directrix is greater in value than the focal point. When this occurs we know that the parabola opens downward. The focal point lies along the y-axis. since the vertex is the midpoint of the line segment from the focal point to the directrix, the vertex is the origin. All of this implies that the parabola is in standard position with axis along the y-axis. This tells us which equation to use. y 1 1 2 2 y x x 4(2) 8 1 2 x 4c Ex 2: Find the focal point and directrix of the parabola with equation 2 y2 8y x 7 0 Solution: Complete the square on the y variable so that we can compare this equation with the equation of a parabola in standard position. 2( y 2 4 y ) 7 x 2( y 2 4 y 4) 2(4) 7 x 2( y 2) 2 1 x This graph is a translation of the equation x = 2y2 We know that this equation will open right or left with axis along or parallel to the x-axis. x 1 2 y 4c If we compare the parent function for this example to the standard form equation for this example … x = 2y2 We see that 1 4c 8c 1 1 c 8 2 1 4c Now solve for c. 2 Focal Point: 1 ,0 8 Since c is positive we know that the graph opens to the right. Directrix: x 1 8 Ex 3: Determine the equation of the parabola with focal point at (1, 5) and directrix y = -1 Solution: From the given information we know that the parabola opens upward. The distance between the focal point and directrix is 6 units. This tells us that the vertex is at (1, 2) since the vertex is midway between the two. This also tells us that c = 3. Therefore, the focal point has coordinate ( 0, 3) We must use equation: 1 2 y x 4c Substitute the value of c into the formula and simplify. 1 2 y x 4(3) 1 y x2 12 Since the vertex is at (1, 2) this tells we have a vertical shift of 2 units and a horizontal shift of 1 unit to the right. 1 2 y x 12 1 y ( x 1) 2 2 12 You may also see this written as: y2 1 ( x 1) 2 12 Parabolas have a distinctive reflective property. Any wave (sound, light, etc.) gets reflected from the focus. Focus Note: If a parabola is rotated about its x-axis, a surface is created called a Paraboloid. Therefore, all of the cross-sections of this paraboloid, using the same axis, share the same focal point. Ex 4: A satellite dish receiver has its amplifier in line with the edge of the dish. The diameter of the dish at the edge is 1 meter. How deep is the dish? y 1 meter 1 2 , c x This figure represents a parabola in standard position with equation: y Cross-section of dish 1 2 x 4c Because the diameter is 1 meter and half of the satellite lies on each side of the y-axis, we know that the point on the parabola is (1/2, c). Since ½ is on the curve, we can substitute ½ into the formula for x and substitute c in for y. 1 2 x 4c 2 1 1 c 4c 2 1 c 16c 1 c2 16 1 c 4 y Therefore, the distance from the focal point and the vertex (depth of the dish) is 0.25 m or 25 centimeters.