Precalculus: Definition and Equation of a Hyperbola PDF

Lesson 4.1 Definition and Equation of a Hyperbola Precalculus Capstone Project Science, Technology, Engineering, and Mathematics The concept of hyperbolas is widely used in navigation and communication. 2 The ship’s location is determined by examining the difference between...

Lesson 4.1 Definition and Equation of a Hyperbola Precalculus Capstone Project Science, Technology, Engineering, and Mathematics The concept of hyperbolas is widely used in navigation and communication. 2 The ship’s location is determined by examining the difference between the times it receives radio signals from fixed land-based navigation transmitters. 3  Such application of hyperbolas makes it an important concept in Mathematics and in other fields. 4  Learning Competencies At the end of the lesson, you should be able to do the following: Define a hyperbola (STEM-PC11AG-ID-1). Determine the standard form of equation of a hyperbola (STEM-PC11AG-ID-2). 5  Learning Objectives At the end of the lesson, you should be able to do the following: Define a hyperbola. Name the parts and properties of a hyperbola. Write the equation of a hyperbola. Transform the standard form of equation of a hyperbola into its general form and vice versa. 6  Hyperbola A hyperbola is formed when a vertical plane intersects a double-napped cone. 7  Hyperbola It is defined as the set of all points on a plane whose absolute difference between the distances from two fixed points 𝐹1 and 𝐹2 is constant. 8  Hyperbola Like an ellipse, each of the two fixed points 𝐹1 and 𝐹2 is called a focus (plural: foci) of the hyperbola. 9  Hyperbola In the given hyperbola on the right, 𝑷𝟏 𝑭𝟏 − 𝑷𝟏 𝑭𝟐 = 𝑷𝟐 𝑭𝟏 − 𝑷𝟐 𝑭𝟐. 10  Parts of a Hyperbola The line passing through the foci of a hyperbola is called the principal axis. 11  Parts of a Hyperbola The two points on the hyperbola that lie on the principal axis are the vertices. 12  Parts of a Hyperbola The line segment joining the vertices is called the transverse axis. 13  Parts of a Hyperbola The midpoint of the transverse axis is the center of the hyperbola. 14  Parts of a Hyperbola The distance from the center to a focus is called the focal distance. 15  Parts of a Hyperbola We use 𝒂 to denote the distance from the center to a vertex, which is also half of the transverse axis. This means that the length of the transverse axis is 𝟐𝒂. 16  Parts of a Hyperbola The focal distance is denoted by 𝒄. This means that the distance between the two foci is 𝟐𝒄. 17 Parts of a Hyperbola Since a focus is farther from the center than a vertex, 𝑐 > 𝑎. This implies that 𝑐 2 − 𝑎2 > 0. 18 Parts of a Hyperbola We can let 𝑏 be a positive number such that 𝑏 2 = 𝑐 2 − 𝑎2. 19  Parts of a Hyperbola A conjugate axis is the line segment perpendicular to the transverse axis whose length is 2𝑏. 20  Parts of a Hyperbola The length of the conjugate axis is illustrated on the right. 21  Parts of a Hyperbola We can show that if 𝑃 is a point on a hyperbola, then by the definition, 𝑷𝑭𝟏 − 𝑷𝑭𝟐 = 𝟐𝒂. 22  Parts of a Hyperbola Suppose 𝑃 is close to 𝐹1. Then 𝑃𝐹1 = 𝑐 and 𝑃𝐹2 = 2𝑎 + 𝑐. Hence, 𝑷𝑭𝟏 − 𝑷𝑭𝟐 = 𝒄 − 𝟐𝒂 + 𝒄 = 𝟐𝒂 23  Let’s Practice! If the focal distance of a hyperbola is 10 units and half of its transverse axis is 8 units, what is the distance from the center of the hyperbola to one of the endpoints of the conjugate axis? 24  Let’s Practice! If the focal distance of a hyperbola is 10 units and half of its transverse axis is 8 units, what is the distance from the center of the hyperbola to one of the endpoints of the conjugate axis? 6 units 25 Try It! Find the distance from the center of a hyperbola to one of its vertices if the focal distance is 13 units and the distance from the center to an endpoint of the conjugate axis is 12 units. 26  Let’s Practice! The foci of a hyperbola are at ±𝟖, 𝟎. Find the distance from the center of the hyperbola to one of its vertices if the endpoints of the conjugate axis are at 𝟎, ±𝟔. 27  Let’s Practice! The foci of a hyperbola are at ±𝟖, 𝟎.Find the distance from the center of the hyperbola to one of its vertices if the endpoints of the conjugate axis are at 𝟎, ±𝟔. 𝟐 𝟕 units 28 Try It! The foci of a hyperbola are at ±𝟏𝟐, 𝟎. Find the distance from the center of the hyperbola to one of its vertices if the endpoints of the conjugate axis are at 𝟎, ±𝟏𝟎. 29  Let’s Practice! The foci of a hyperbola are at (±𝟒, 𝟎). Find the distance from the center of the hyperbola to one of its vertices if 𝑷(−𝟓, 𝟑) is a point on the graph. 30  Let’s Practice! The foci of a hyperbola are at (±𝟒, 𝟎). Find the distance from the center of the hyperbola to one of its vertices if 𝑷(−𝟓, 𝟑) is a point on the graph. 𝟏𝟎 units 31 Try It! The foci of a hyperbola are at (±𝟑, 𝟎). Find the distance from the center of the hyperbola to one of its vertices if 𝑷(−𝟓, 𝟒) is a point on the graph. 32 How do you represent the equation of a hyperbola? 33  Equation of a Hyperbola in Standard Form The standard form of the equation of a hyperbola with center at the origin and transverse axis on the 𝒙-axis is 𝒙𝟐 𝒚𝟐 𝟐 − 𝟐 = 𝟏. 𝒂 𝒃 34  Equation of a Hyperbola in Standard Form If the center is at the origin and the transverse axis is on the 𝒚-axis, then the equation of the hyperbola becomes 𝒚𝟐 𝒙𝟐 𝟐 − 𝟐 = 𝟏. 𝒂 𝒃 35  Equation of a Hyperbola in Standard Form If the center is at 𝒉, 𝒌 and the transverse axis is horizontal, then the equation of the hyperbola is 𝟐 𝟐 𝒙−𝒉 𝒚−𝒌 − = 𝟏. 𝒂𝟐 𝒃𝟐 36  Equation of a Hyperbola in Standard Form Similarly, if the center is at 𝒉, 𝒌 and the transverse axis is vertical, then the equation of the hyperbola is 𝟐 𝟐 𝒚−𝒌 𝒙−𝒉 − = 𝟏. 𝒂𝟐 𝒃𝟐 37  Let’s Practice! What is the standard form of equation of a hyperbola with center at the origin, transverse axis on the 𝒙-axis, 𝒂 = 𝟓, and 𝒃 = 𝟑? 38  Let’s Practice! What is the standard form of equation of a hyperbola with center at the origin, transverse axis on the 𝒙-axis, 𝒂 = 𝟓, and 𝒃 = 𝟑? 𝒙𝟐 𝒚𝟐 − =𝟏 𝟐𝟓 𝟗 39 Try It! Find the standard form of equation of a hyperbola with center at the origin, transverse axis on the 𝒚-axis, 𝒂 = 𝟖, and 𝒃 = 𝟔. 40  Let’s Practice! Determine the standard form of equation of a hyperbola whose center is at (𝟐, −𝟏), transverse axis is vertical, 𝒃 = 𝟖, and 𝒄 = 𝟏𝟎. 41  Let’s Practice! Determine the standard form of equation of a hyperbola whose center is at (𝟐, −𝟏), transverse axis is vertical, 𝒃 = 𝟖, and 𝒄 = 𝟏𝟎. 𝒚+𝟏 𝟐 𝒙−𝟐 𝟐 − =𝟏 𝟑𝟔 𝟔𝟒 42 Try It! Determine the standard form of equation of a hyperbola whose center is at (−𝟒, 𝟑), transverse axis is horizontal, 𝒂 = 𝟗, and 𝒄 = 𝟏𝟔. 43  Let’s Practice! Determine the standard form of the equation of a hyperbola with vertices at (𝟑, 𝟏) and (𝟏𝟏, 𝟏) and foci at (−𝟑, 𝟏) and (𝟏𝟕, 𝟏). 44  Let’s Practice! Determine the standard form of the equation of a hyperbola with vertices at (𝟑, 𝟏) and (𝟏𝟏, 𝟏) and foci at (−𝟑, 𝟏) and (𝟏𝟕, 𝟏). 𝒙−𝟕 𝟐 𝒚−𝟏 𝟐 − =𝟏 𝟏𝟔 𝟖𝟒 45 Try It! Determine the standard form of equation of a hyperbola with vertices at (𝟔, 𝟏) and (−𝟐, 𝟏) and endpoints of the conjugate axis at (𝟐, −𝟒) and (𝟐, 𝟔). 46 What values are needed to write the standard form of the equation of a hyperbola? 47  Equation of a Hyperbola in General Form The general form of equation of a hyperbola is represented by 𝑨𝒙𝟐 + 𝑩𝒚𝟐 + 𝑪𝒙 + 𝑫𝒚 + 𝑬 = 𝟎 where 𝐴 and 𝐶 are not equal to 0. 48  Equation of a Hyperbola in General Form Note that the coefficients 𝑥 2 and 𝑦 2 have different signs because of the subtraction of the terms in the standard form. Hence, in the general form of equation of a hyperbola, 𝐴𝐵 < 0. 49  Let’s Practice! Transform the standard form of equation of a 𝒙𝟐 𝒚𝟐 hyperbola given by − = 𝟏 into general form. 𝟏𝟎 𝟏𝟐 50  Let’s Practice! Transform the standard form of equation of a 𝒙𝟐 𝒚𝟐 hyperbola given by − = 𝟏 into general form. 𝟏𝟎 𝟏𝟐 𝟔𝒙𝟐 − 𝟓𝒚𝟐 − 𝟔𝟎 = 𝟎 51 Try It! Given a hyperbola, transform its 𝒙𝟐 𝒚𝟐 standard form of equation − =𝟏 𝟑𝟎 𝟐𝟎 into general form. 52  Let’s Practice! Determine the general form of equation of a 𝒚−𝟓 𝟐 𝒙+𝟏 𝟐 hyperbola whose standard form is − = 𝟏. 𝟏𝟔 𝟗 53  Let’s Practice! Determine the general form of equation of a 𝒚−𝟓 𝟐 𝒙+𝟏 𝟐 hyperbola whose standard form is − = 𝟏. 𝟏𝟔 𝟗 𝟗𝒚𝟐 − 𝟏𝟔𝒙𝟐 − 𝟗𝟎𝒚 − 𝟑𝟐𝒙 + 𝟔𝟓 = 𝟎 54 Try It! Determine the general form of equation of a hyperbola whose standard form is 𝒙+𝟒 𝟐 𝒚−𝟑 𝟐 − = 𝟏. 𝟐𝟎 𝟗 55  Let’s Practice! In the given figure, Stations A and B are 100 kilometers apart. Suppose ship 𝑷 receives Station A’s signal 240 microseconds before it receives the signal from Station B. Write an equation in general form for the branches indicated in the figure. Assume that signal travels at 0.3 kilometers per microsecond. 56  Let’s Practice! In the given figure, Stations A and B are 100 kilometers apart. Suppose ship 𝑷 receives Station A’s signal 240 microseconds before it receives the signal from Station B. Write an equation in general form for the branches indicated in the figure. Assume that signal travels at 0.3 kilometers per microsecond. 𝟑𝟎𝟏𝒙𝟐 − 𝟔𝟐𝟓𝒚𝟐 − 𝟕𝟓𝟐 𝟓𝟎𝟎 = 𝟎 57 Try It! Two stations that are 8 km apart receive a sound signal from a source located at a certain point. The station located at (𝟖, 𝟎) receives the signal 20 seconds earlier than the station located at (𝟎, 𝟎). Use 0.3 km/sec as the speed of sound and find an equation in standard form containing the source of the sound. 58  Check Your Understanding Determine the measure indicated in each item by using the relationship between 𝒂, 𝒃, and 𝒄 in a hyperbola. 1. Determine the value of 𝑎 if 𝑐 = 13 and 𝑏 = 12. 2. Determine the value of 𝑐 if 𝑎 = 10 and 𝑏 = 15. 3. Determine the focal distance of a hyperbola if the transverse axis is 8 units long and the conjugate axis is 6 units long. 59  Check Your Understanding Write the standard and general forms of equation of the hyperbola satisfying the given conditions. 1. center at the origin, vertices at (±8, 0), and endpoints on the conjugate axis at (0, ±6) 2. center at the origin, vertices at (±3, 0), and foci at (±5, 0) 3. one vertex at (3,1) and an endpoint of the vertical conjugate axis at 0,4 60  Let’s Sum It Up! A hyperbola is formed when a vertical plane intersects a double-napped cone. A hyperbola is a set of points in a plane whose absolute value of the difference between the distances from two fixed points is constant. 61  Let’s Sum It Up! The two fixed points in the definition of a hyperbola are the foci (singular: focus). The line passing through the foci of a hyperbola is called the principal axis. The two points on a hyperbola that lie on the principal axis are the vertices. 62  Let’s Sum It Up! The line segment joining the vertices of a hyperbola is called the transverse axis. The midpoint of the transverse axis is the center of the hyperbola. 63  Let’s Sum It Up! The line segment that passes through the center and is perpendicular to the transverse axis whose length is 2𝑏 is called the conjugate axis. The distance from the center to a focus of a hyperbola is called the focal distance. 64  Key Formulas Concept Formula Description Use this formula to find Equation of a 𝑥−ℎ 2 𝑦−𝑘 2 − = 1, the equation of a Hyperbola in 𝑎2 𝑏2 hyperbola given its Standard From where center, 𝑎, and 𝑏 if the (ℎ, 𝑘) is the center, transverse axis is 𝑎 is the distance from horizontal. the center to a vertex, and 𝑏 is the distance from the center to an endpoint of the conjugate axis. 65  Key Formulas Concept Formula Description Use this formula to find Equation of a 𝑦−𝑘 2 𝑥−ℎ 2 − = 1, the equation of a Hyperbola in 𝑎2 𝑏2 hyperbola given its Standard From where center, 𝑎, and 𝑏 if the (ℎ, 𝑘) is the center, transverse axis is 𝑎 is the distance from vertical. the center to a vertex, and 𝑏 is the distance from the center to an endpoint of the conjugate axis. 66  Key Formulas Concept Formula Description This is the equation of Equation of a 𝐴𝑥 2 + 𝐵𝑦 2 + 𝐶𝑥 + 𝐷𝑦 + 𝐸 = 0 a hyperbola when the Hyperbola in standard form is General From expanded. 67 Challenge Yourself The general form of equation of a hyperbola is given by 𝑨𝒙𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 ,where 𝑨𝑪 < 𝟎. Why can’t 𝑨𝑪 be equal to zero or greater than zero? 68 Bibliography Barnett, Raymond, Michael Ziegler, Karl Byleen, and David Sobecki. College Algebra with Trigonometry. Boston: McGraw Hill Higher Education, 2008. Bittinger, Marvin L., Judith A. Beecher, David J. Ellenbogen, and Judith A. Penna. Algebra and Trigonometry: Graphs and Models. 4th ed. Boston: Pearson/Addison Wesley, 2009. Blitzer, Robert. Algebra and Trigonometry. 3rd ed. Upper Saddle River, New Jersey: Pearson/Prentice Hal, 2007. Larson, Ron. College Algebra with Applications for Business and the Life Sciences. Boston: MA: Houghton Mifflin, 2009. Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York: McGraw-Hill, 1996. 69