Download Pre calculus senior high and more Summaries Mathematics in PDF only on Docsity! UNIFIED SCHOOLS OF THE ARCHDIOCESE OF LIPA SAN GUILLERMO ACADEMY TALISAY, BATANGAS MODULE IN PRE-CALCULUS AY 2021- 2022 1 Name: (Surname, First Name Middle Initial) Grade & Section: TITLE Module No. 4: CONIC SECTIONS Lesson No. 5: HYPERBOLA OVERVIEW IT IS EASY TO SEE how of the four basic conics, the hyperbola stands out. For one, it is the only conic section whose graph is not continuous throughout because it is composed of two symmetrical, unbounded branches. Also, while it resembles the parabola, the hyperbola opens toward two opposite directions. While there are not as many structures that take inspiration from the hyperbola as compared to the other conics, its interesting shape is not uncommon in the arts. As example, the National Museum of Natural History incorporates modern geometric architecture and welcomes guests with a splendid structure of a hyperbolic panel that stretches from the ground level to the ceiling and functions as an elevator. In this lesson we ask: How does the hyperbola form its unique shape and what mathematical ideas can be used to describe it? OBJECTIVES โ Define a hyperbola. โ Determine the standard form of equation of a hyperbola. DISCUSSION OF CONTENT The Hyperbola as a Conic Section Visualize the intersection of a plane and a double right circular cone such that a perpendicular plane to the bases of the cone intersects both nappes but not along the vertex of the cone. This intersection is a hyperbola โa conic section composed of two symmetrical, disjoint curves that extend infinitely to opposite directions. Graphically, the hyperbola is defined by two fixed points in its interior called its foci. Suppose the difference between the distances from a point ๐(๐ฅ,๐ฆ) to two fixed points is equal to some constant. This point, together with the set of all other points G11- St. UNIFIED SCHOOLS OF THE ARCHDIOCESE OF LIPA SAN GUILLERMO ACADEMY TALISAY, BATANGAS MODULE IN PRE-CALCULUS AY 2021- 2022 2 whose differences in distances to the foci are the same constant form a hyperbola. Like the other conics we have studied, the hyperbola has essential features or properties in its graph. We take note of the following: a) Vertex marks the point where the graph changes direction. The midpoint of the two vertices is a point called the centre of the hyperbola. b) The foci are points of the hyperbola which are collinear to the centre and the vertices. The shape of a hyperbola is determined by several pairs of lines and segments. c) Asymptotes are the two lines intersecting at the center. d) Auxiliary rectangles are rectangles whose diagonals pass through the centre and are subsets of the asymptotes. e) The midpoint of each of the horizontal sides of the rectangle is a co-vertex of the hyperbola. f) Transverse axis-the shortest segment connecting two points from different branches of the hyperbola. The length of the transverse axis is 2๐ which happens to be the length of the auxiliary rectangle. g) Conjugate axis is perpendicular to the transverse axis and connecting the co- vertices. Based on the coordinates of the co-vertices, the length of the conjugate axis and the width of the auxiliary rectangle is 2๐.
