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 M athematics and in other fields. 4 Learning Objectives At the end of the lesson, you should be able to do the following: β Define a hyperbola. β Name the parts and properties of a hyperbola. β Write the equation of a hyperbola. β Transform the standard form of equation of a hyperbola into its general form and vice versa. 5 Hyperbola A hyperbola is formed when a vertical plane intersects a double-napped cone. 6 Hyperbola It is defined as the set of all points on a plane whose absolute difference between the distances from two fixed points πΉ1 and πΉ2 is constant. 7 Hyperbola Like an ellipse, each of the two fixed points πΉ1 and πΉ2 is called a focus (plural: foci) of the hyperbola. 8 Hyperbola In the given hyperbola on the right, π·π ππ − π·π ππ = π·π ππ − π·π ππ . 9 Parts of a Hyperbola The line passing through the foci of a hyperbola is called the principal axis. 10 Parts of a Hyperbola The two points on the hyperbola that lie on the principal axis are the vertices. 11 Parts of a Hyperbola The line segment joining the vertices is called the transverse axis. 12 Parts of a Hyperbola The midpoint of the transverse axis is the center of the hyperbola. 13 Parts of a Hyperbola The distance from the center to a focus is called the focal distance. 14 Parts of a Hyperbola We use π 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 ππ. 15 Parts of a Hyperbola The focal distance is denoted by π. This means that the distance between the two foci is ππ. 16 Parts of a Hyperbola Since a focus is farther from the center than a vertex, π > π. This implies that π 2 − π2 > 0. 17 Parts of a Hyperbola We can let π be a positive number such that π 2 = π 2 − π2 . 18 Parts of a Hyperbola A conjugate axis is the line segment perpendicular to the transverse axis whose length is 2π. 19 Parts of a Hyperbola The length of the conjugate axis is illustrated on the right. 20 Parts of a Hyperbola We can show that if π is a point on a hyperbola, then by the definition, π·ππ − π·ππ = ππ. 21 Parts of a Hyperbola Suppose π is close to πΉ1 . Then ππΉ1 = π and ππΉ2 = 2π + π. Hence, π·ππ − π·ππ = π − ππ + π = ππ 22 Let’s Practice! If the focal distance of a hyperbola is 1 0 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? 23 Let’s Practice! If the focal distance of a hyperbola is 1 0 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 24 Let’s Practice! The foci of a hyperbola are at ±π, π . Find the distance from the center of the hyperbola to one of its vertices if the endpoints of the conjugate axis are at π, ±π . 25 Let’s Practice! The foci of a hyperbola are at ±π, π .Find the distance from the center of the hyperbola to one of its vertices if the endpoints of the conjugate axis are at π, ±π . π π units 26 Let’s Practice! The foci of a hyperbola are at (±π, π). Find the distance from the center of the hyperbola to one of its vertices if π·(−π, π) is a point on the graph. 27 Let’s Practice! The foci of a hyperbola are at (±π, π). Find the distance from the center of the hyperbola to one of its vertices if π·(−π, π) is a point on the graph. ππ units 28 How do you represent the equation of a hyperbola? 29 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 π-axis is ππ ππ − π = π. π π π 30 Equation of a Hyperbola in Standard Form If the center is at the origin and the transverse axis is on the π-axis, then the equation of the hyperbola becomes ππ ππ − π = π. π π π 31 Equation of a Hyperbola in Standard Form If the center is at π, π and the transverse axis is horizontal, then the equation of the hyperbola is π−π π π−π π − = π. π π π π 32 Equation of a Hyperbola in Standard Form Similarly, if the center is at π, π and the transverse axis is vertical, then the equation of the hyperbola is π−π π π−π π − = π. π π π π 33 Let’s Practice! What is the standard form of equation of a hyperbola with center at the origin, transverse axis on the π-axis, π = π, and π = π? 34 Let’s Practice! What is the standard form of equation of a hyperbola with center at the origin, transverse axis on the π-axis, π = π, and π = π? ππ ππ − =π ππ π 35 Let’s Practice! Determine the standard form of equation of a hyperbola whose center is at (π, −π), transverse axis is vertical, π = π, and π = ππ. 36 Let’s Practice! Determine the standard form of equation of a hyperbola whose center is at (π, −π), transverse axis is vertical, π = π, and π = ππ. π+π π π−π π − =π ππ ππ 37 Let’s Practice! Determine the standard form of the equation of a hyperbola with vertices at (π, π) and (ππ, π) and foci at (−π, π) and (ππ, π). 38 Let’s Practice! Determine the standard form of the equation of a hyperbola with vertices at (π, π) and (ππ, π) and foci at (−π, π) and (ππ, π). π−π π π−π π − =π ππ ππ 39 Equation of a Hyperbola in General Form The general form of equation of a hyperbola is represented by π¨ππ + π©ππ + πͺπ + π«π + π¬ = π where π΄ and πΆ are not equal to 0. 40 Equation of a Hyperbola in General Form Note that the coefficients π₯ 2 and π¦ 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, π΄π΅ < 0. 41 Let’s Practice! Transform the standard form of equation of a ππ ππ hyperbola given by − = π into general form. ππ ππ 42 Let’s Practice! Transform the standard form of equation of a ππ ππ hyperbola given by − = π into general form. ππ ππ πππ − πππ − ππ = π 43 Let’s Practice! Determine the general form of equation of a hyperbola whose standard form is π−π π π+π π − = π. ππ π 44 Let’s Practice! Determine the general form of equation of a hyperbola whose standard form is π−π π π+π π − = π. ππ π πππ − ππππ − πππ − πππ + ππ = π 45 Let’s Sum It Up! β A hyperbola is formed when a vertical plane intersects a double-napped cone. β 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. 46 Let’s Sum It Up! β The two fixed points in the definition of a hyperbola are the foci (singular: focus). β The line passing through the foci of a hyperbola is called the principal axis. β The two points on a hyperbola that lie on the principal axis are the vertices. 47 Let’s Sum It Up! β The line segment joining the vertices of a hyperbola is called the transverse axis. β The midpoint of the transverse axis is the center of the hyperbola. 48 Let’s Sum It Up! β The line segment that passes through the center and is perpendicular to the transverse axis whose length is 2π is called the conjugate axis. β The distance from the center to a focus of a hyperbola is called the focal distance. 49 Key Formulas C oncept E quation of a Hyperbola in Standard From Formula π₯−β 2 π¦−π 2 − = 1, π2 π2 where (β, π) is the center, π is the distance from the center to a vertex, and π is the distance from the center to an endpoint of the conjugate axis. Description U se this formula to find the equation of a hyperbola given its center, π, and π if the transverse axis is horizontal. 50 Key Formulas C oncept E quation of a Hyperbola in Standard From Formula π¦−π 2 π₯−β 2 − = 1, π2 π2 where (β, π) is the center, π is the distance from the center to a vertex, and π is the distance from the center to an endpoint of the conjugate axis. Description U se this formula to find the equation of a hyperbola given its center, π, and π if the transverse axis is vertical. 51 Key Formulas C oncept E quation of a Hyperbola in G eneral From Formula π΄π₯ 2 + π΅π¦ 2 + πΆπ₯ + π·π¦ + πΈ = 0 Description This is the equation of a hyperbola when the standard form is expanded. 52 Challenge Yourself The general form of equation of a hyperbola is given by π¨ππ + πͺππ + π«π + π¬π + π = π ,where π¨πͺ < π. Why can’t π¨πͺ be equal to zero or greater than zero? 53
