MATH 110 analytic geometry Conics. The Parabola

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1 1 MATH 11 analytic geometry Conics The graph of a second-degree equation in the coordinates x and y is called a conic section or, more simply, a conic. This designation derives from the fact that the curve can be obtained as the intersection of a right circular cone and a plane. The general quadratic equation in the equation in x and y may be expressed in the form Ax Bxy Cy Dx Ey F The Greek mathematician Apollonius (6 BC BC) wrote the definitive treatise Conic Sections on this subject. It superseded the works of earlier Greek geometers and formed the cornerstone of thought on the subject for well over a thousand years. Indeed, eighteen centuries passed before Descartes wrote his La Géométrie. The Parabola A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed line of the plane. The fixed point is called the focus and the fixed line is called the directrix. Elements of Parabola Focus point located at distance a from the vertex in the direction of parabola s opening. Directrix a straight line located at distance a from the vertex and at the opposite side of the parabola s opening. Vertex the point extremity of parabola, i.e. highest point for parabolas opening downward, lowest point for parabolas opening upward, rightmost point for parabolas opening leftward, and leftmost point for parabolas opening rightward. The coordinates of the vertex is denoted by ( hk, ). Axis the line of symmetry of parabola. It contains bothe the focus and the vertex and always perpendicular to the directrix. Latus Rectum denoted by LR, is a line perpendicular to the axis, passing through the focus and terminates on the parabola itself. Eccentricity a measure of how much a conic deviate from being circular, making the eccentricity of the circle obviously equal to zero. It is the ration of focal distance to directix distance of the conic section. The eccentricity of parabola is always 1 ( e 1). Thus, parabola can also be defined as a conic section of eccentricity 1. Fuller, G., & Tarwater D. (199). Analytic Geometry. 7 th Edition, Addison Wesley Publishing Company, Inc.

2 Although points of a parabola can be located by a direct application of the definition of a parabola, it is easier to obtain them from an equation of the curve. The simplest equation of a parabola can be written if the coordinate axes are placed in special position relative to the directrix and focus. Let the x - axis be on the line through the focus and perpendicular to the directrix and let the vertex be on the origin. Then, choosing a, we denote the coordinates of the focus by Fa (,), and the equation of the directrix by x 1. Since any point P( x, y ) of the parabola is the same distance from the focus and directrix, we have terms. Thus, ( ) ( ) x a y x a We next square the binomials in this equation and collect ( xa) y ( x a) x ax a y x ax a y 4 ax This is the equation of a parabola with the vertex at the origin and focus at ( a,). Since a, x may have positive value or zero, but no negative value. Hence, the graph extends indefinitely far into the first and fourth quadrants, and the axis of the parabola is the positive x - axis. It is evident from the equation that the parabola is symmetric with respect to its axis because y ax. For the latus rectum, its length can be determined from the coordinates of its endpoints. By substituting a for x in the equation y y 4 ax, we find 4a and y a Hence the endpoints are ( a, a) and ( a, ). a This makes the length of the latus rectum equal to 4. a The vertex and the extremities of the latus rectum are sufficient for drawing a rough sketch of the parabola. Theorem. The equation of a parabola with vertex at the origin and focus at is The parabola opens to the right if and opens to the left if The equation of a parabola with vertex at the origin and focus at is The parabola opens upward if and opens downward if Example 1. Write the equation of the parabola with vertex at the origin and the focus at (,4). Graph the parabola. Fuller, G., & Tarwater D. (199). Analytic Geometry. 7 th Edition, Addison Wesley Publishing Company, Inc.

3 3 Example. A parabola has its vertex at the origin, its axis along the x - axis, passes through the point ( 3,6). Find its equation. Example 3. The equation of a parabola is the directrix, and the length of the latus rectum. x 6 y. Find the coordinates of the focus, the equation of Example 4. A cable suspended from supports that are the same height and 6 feet apart has a sag of 1 feet. If the cable hangs in the form of a parabola, find its equation, taking the origin at the lowest point. Exercises. A. Find the coordinates of the focus, the length of the latus rectum, and the coordinates of its endpoints for each of the given parabolas. Find also the equation of the directrix of each parabola. Sketch the curve. 1. y 16x 3. y 3 x Fuller, G., & Tarwater D. (199). Analytic Geometry. 7 th Edition, Addison Wesley Publishing Company, Inc.

4 4. x 1y 4. x 8y B. Write the equation of the parabola with vertex at the origin that satisfies the given conditions. 5. Focus at (3,) 7. Directrix is x 4 6. Focus at (, 4) 8. Directrix is y 7 9. The length of the latus rectum is 1 and the parabola opens to the right. 1. The focus is on the x - axis and the parabola passes through the point ( 3,4). Fuller, G., & Tarwater D. (199). Analytic Geometry. 7 th Edition, Addison Wesley Publishing Company, Inc.

5 5 Parabola with Vertex at ( hk, ) Theorem. The equation of a parabola with vertex at and focus at is The parabola opens to the right if and opens to the left if The equation of a parabola with vertex at and focus at is The parabola opens upward if and opens downward if Each of the above equations is said to be in standard form. When h and k, they reduce to the simpler equations of the preceding section. If the equation of the parabola is in standard form, its graph can be quickly sketched. The vertex and the ends of the latus rectum are sufficient for a rough sketch. The plotting of a few additional points would, of course, improve the accuracy. We note that each of the equations is quadratic in one variable and linear in the other variable. This fact can be expressed more vividly if we perform the indicated squares and transpose to obtain the general forms x Dx Ey F and y Dx Ey F Conversely, an equation in general form can be presented in a standard form, provided that E in the first general equation and D in the second general equation. Example 1. Draw the graph of the equation y x y Example. Construct the graph of the equation x x y Fuller, G., & Tarwater D. (199). Analytic Geometry. 7 th Edition, Addison Wesley Publishing Company, Inc.

6 6 Example 3. A parabola whose axis is parallel to the y - axis passes through the points (1,1), (,), and ( 1,5). Find its equation and sketch the graph. Exercises. A. Express the equation, in standard form, of the parabola that satisfies the given condition. 1. Vertex at (3,), focus at (3,4) 3. Vertex at (4,1), x as directrix. Vertex at ( 6, 4), focus at (, 4) 4. Vertex at (4, ), y 3 as directrix 5. Vertex at (4, ), latus rectum 8 ; opens to the right 6. Vertex at (3, ), ends of latus rectum 1,, 1 8,. Fuller, G., & Tarwater D. (199). Analytic Geometry. 7 th Edition, Addison Wesley Publishing Company, Inc.

7 7 7. Focus at (5,), x 6 as directrix. 8. Focus at (,), y 4 as directrix. B. Express each equation in standard form. Give the coordinates of the vertex, the focus, and the ends of the latus rectum. Sketch the graph. 9. y 8x8 11. x 1x y 5 1. x 4x 16y 4 1. y 1x 6y 1 C. Find the equation of the parabola satisfying the following conditions: 13. Vertex at (3, 4), axis horizontal; passes through (, 5). 14. Vertex at ( 1, ), axis vertical; passes through (3,6). 15. Axis vertical; passes through ( 1,), (5,), and (1,8). Fuller, G., & Tarwater D. (199). Analytic Geometry. 7 th Edition, Addison Wesley Publishing Company, Inc.