CONIC SECTIONS Prepared by: Prof. Teresita P. Liwanag – ZapantaB.S.C.E., M.S.C.M., M.Ed. (Math-units), PhD-TM (on-going)
SPECIFIC OBJECTIVES: At the end of the lesson, the student is expected tobe able to:• define conic section• identify the different conic section• describe parabola• convert general form to standard form of equation of parabola and vice versa.• give the different properties of a parabola and sketch itsgraph
Conic Section or a Conic is a path of point that moves sothat its distance from a fixed point is in constant ratio to itsdistance from a fixed line. Focus is the fixed point Directrix is the fixed line Eccentricity is the constant ratio usually represented by (e)
The conic section falls into three (3) classes, which varies inform and in certain properties. These classes are distinguishedby the value of the eccentricity (e). If e = 1, a conic section which is a parabola If e < 1, a conic section which is an ellipse If e > 1, a conic section which is a hyperbola
THE PARABOLA (e = 1) A parabola is the set of all points in a plane, which areequidistant from a fixed point and a fixed line of the plane.The fixed point called the focus (F) and the fixed line thedirectrix (D). The point midway between the focus and thedirectrix is called the vertex (V). The chord drawn throughthe focus and perpendicular to the axis of the parabola iscalled the latus rectum (LR).
Equations of parabola with vertex at the origin V (0, 0)
Examples1. Determine the focus, the length of the latus rectum andthe equation of the directrix for the parabola 3y2 – 8x = 0 andsketch the graph.2. Write the equation of the parabola with vertex V at (0, 0)which satisfies the given conditions:a. axis on the y-axis and passes through (6, -3)b. F(0, 4/3) and the equation of the directrix is y + 4/3 = 0c. Directrix is x – 4 = 0d. Focus at (0, 2)e. Latus rectum is 6 units and the parabola opens to the leftf. Focus on the x-axis and passes through (4, 3)
We consider a parabola whose axis is parallel to, butnot on, a coordinate axis. In the figure, the vertex is at (h, k)and the focus at (h+a, k). We introduce another pair of axes bya translation to the point (h, k). Since the distance from thevertex to the focus is a, we have at once the equation y’2 = 4ax’ Therefore the equation of a parabola with vertex at (h,k) and focus at (h+a, k) is (y – k)2 = 4a (x – h)
Standard Form General Form(y – k)2 = 4a (x – h) y2 + Dy + Ex + F = 0(y – k)2 = - 4a (x – h)(x – h)2 = 4a (y – k) x2 + Dx + Ey + F = 0(x – h)2 = - 4a (y – k)
Examples1. Draw the graph of the parabola y2 + 8x – 6y + 25 = 02. Express x2 – 12x + 16y – 60 = 0 to standard form and constructthe parabola.3. Determine the equation of the parabola in the standard form,which satisfies the given conditions.a. V (3, 2) and F (5, 2)b. V (2, 3) and axis parallel to y axis and passing through (4, 5)c. V (2, 1), Latus rectum at (-1, -5) & (-1, 7)d. V (2, -3) and directrix is y = -74. Find the equation of parabola with vertex at (-1, -2), axis isvertical and passes through (3, 6).
5. A parabola whose axis is parallel to the y-axis passes through thepoints (1, 1), (2, 2) and (-1, 5). Find the equation and construct theparabola.6. A parabola whose axis is parallel to the x-axis passes through(0, 4), (0, -1) and (6, 1). Find the equation and construct theparabola.7. A parabolic trough 10 meters long, 4 meters wide across the topand 3 meters deep is filled with water at a depth of 2 meters. Findthe volume of water in the trough.8. Water spouts from a horizontal pipe 12 meters above theground and 3 meters below the line of the pipe, the watertrajectory is at a horizontal distance of 5 meters. How far from thevertical line will the stream of the water hit the ground?
9. A parabolic suspension bridge cable is hung between twosupporting towers 120 meters apart and 35 meters above thebridge deck. The lowest point of the cable is 5 meters above thedeck. Determine the lengths (h1 & h2) of the tension members 20meters and 40 meters from the bridge center.10. A parkway 20 meters wide is spanned by a parabolic arc 30meters long along the horizontal. If the parkway is centered, howhigh must the vertex of the arch be in order to give a minimumclearance of 5 meters over the parkway.