The document provides the step-by-step solution to solving a hyperbola equation in standard form. It gives the center, vertices, foci, transverse and conjugate axes, and asymptotes. It then solves two word problems involving hyperbolas, determining the distance between houses shaped as branches of a hyperbola based on the given equation.
This is a powerpoint presentation that discusses about the topic or lesson: Conic Sections. It also includes the definition, types and some terminologies involved in the topic: Conic Sections.
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
This is a powerpoint presentation that discusses about the topic or lesson: Conic Sections. It also includes the definition, types and some terminologies involved in the topic: Conic Sections.
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
21 - GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES.pptxbernadethvillanueva1
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
21 - GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES.pptxbernadethvillanueva1
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
Pre-Calculus: Conics - Introduction to Conics and Determining & Graphing Circ...Myrrhtaire Castillo
This PowerPoint contains an introduction to conical sections: the conics formed from double-napped circular cone - the Parabola, Hyperbola, Circle, & Ellipse. It also contains the basic parts of Circle. Identifying the standard form of circle's radius and center. Graphing a circle from its standard form. Transforming General Equation of Circle to Standard Form and some of the special cases.
x2 y2
Standard Equation of hyperbola is a 2 – b2 = 1
(i) Definition hyperbola : A Hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed line (called directrix) is always constant which is always greater than unity.
The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called conjugate hyperbola.
Note :
(i) If e1 and e2 are the eccentricities of the
(ii) Vertices : The point A and A where the curve meets the line joining the foci S and S
hyperbola and its conjugate then
1 +
e 2 e
1 = 1
2
are called vertices of hyperbola.
(iii) Transverse and Conjugate axes : The straight line joining the vertices A and A is called transverse axes of the hyperbola. Straight line perpendicular to the transverse axes and passes through its centre called conjugate axes.
(iv) Latus Rectum : The chord of the hyperbola which passes through the focus and is perpendicular to its transverse axes is called
2b2
latus rectum. Length of latus rectum = a .
(ii) The focus of hyperbola and its1 conju2gate are concyclic.
Standard Equation and Difinitions
Ex.1 Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1,2) and
eccentricity 3 .
Sol. Let P (x,y) be any point on the hyperbola. Draw PM perpendicular from P on the directrix.
Then by definition SP = e PM
(v) Eccentricity : For the hyperbola
x2 y2
a 2 – b2
= 1,
(SP)2 = e2(PM)2
2x y 12
b2 = a2 (e2 – 1)
(x–1)2 + (y–2)2 = 3
Conjugate axes 2
5(x2 + y2 – 2x – 4y + 5} =
e = =
1
Transverse
axes
3(4x2 + y2 + 1+ 4xy – 2y – 4x)
7x2 – 2y2 + 12xy – 2x + 14y – 22 = 0
(vi) Focal distance : The distance of any point on the hyperbola from the focus is called the focal distance of the point.
Note : The difference of the focal distance of a point on the hyperbola is constant and is equal to the length
of the transverse axes. |SP – SP| = 2a (const.)
which is the required hyperbola.
Ex.2 Find the lengths of transverse axis and conjugate axis, eccentricity and the co- ordinates of foci and vertices; lengths of the latus rectum, equations of the directrices of the hyperbola 16x2 – 9y2 = –144
Sol. The equation 16x2 – 9y2 = – 144 can be
Sol. y= m1(x –a),y= m2(x + a) where m1m2 = k, given
x 2
written as 9
x2
y 2
– 16 = – 1. This is of the form
y2
In order to find the locus of their point of intersection we have to eliminate the unknown
m1 and m2. Multiplying, we get
y2 = m1m2 (x2 – a2) or y2 = k(x2–a2)
a 2 – b2 = – 1
a2 = 9, b2 = 16 a = 3, b = 4
or x – y
1 k
= a2
which represents a hyperbola.
Length of transverse axis :
The length of transverse axis = 2b = 8
Length of conjugate axis :
The length of conjugate axis = 2a = 6
5
Ex.5 T
Pre-calculus 1, 2 and Calculus I (exam notes)William Faber
Notes I typed using Microsoft Word for pre-calculus and calculus exams. Most of the images were also created by me. I shared them with other students in my class to increase their chance of success as well. Upon completion of the courses I donated them to the math center to help other math students.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
3. 1. Express the equation 𝒙 𝟐 − 𝟐𝒚 𝟐 − 𝟐𝒙 − 𝟖𝒚 − 𝟐𝟕 = 𝟎 of the hyperbola
in standard form and solve for the following:
a. Center
b. Vertices
c. Foci
d. Equation of Transverse Axis
e. Length of the Transverse Axis
f. Equation of Conjugate Axis
g. Length of the Conjugate Axis
h. asymptotes
4. Solution: In order to write the equation in standard form, consider first
which coefficient (between x2 and y2) has a negative value then arrange
them in such a way that the first term must be a positive value.
𝒙 𝟐
− 𝟐𝒚 𝟐
− 𝟐𝒙 − 𝟖𝒚 − 𝟐𝟕 = 𝟎
5. 𝒙 𝟐
− 𝟐𝒚 𝟐
− 𝟐𝒙 − 𝟖𝒚 − 𝟐𝟕 = 𝟎
First Step: Arrange the order by grouping like variables
𝒙 𝟐
− 𝟐𝒙 − 𝟐𝒚 𝟐
− 𝟖𝒚 = 𝟐𝟕
6. 𝒙 𝟐
− 𝟐𝒙 − 𝟐𝒚 𝟐
− 𝟖𝒚 = 𝟐𝟕
Second Step: Factor out the coefficient
(𝒙 𝟐
−𝟐𝒙) − 𝟐(𝒚 𝟐
+ 𝟒𝒚) = 𝟐𝟕
9. (𝒙 − 𝟏) 𝟐
−𝟐 𝒚 + 𝟐 𝟐
= 𝟐𝟎
Fifth Step: Divide both side by 20 to make the constant equal to 1
(𝒙 − 𝟏) 𝟐
−𝟐 𝒚 + 𝟐 𝟐
𝟐𝟎
=
𝟐𝟎
𝟐𝟎
10. (𝒙 − 𝟏) 𝟐
−𝟐 𝒚 + 𝟐 𝟐
𝟐𝟎
=
𝟐𝟎
𝟐𝟎
Sixth Step: Simplify the fraction, then you got the standard form.
(𝒙 − 𝟏) 𝟐
𝟐𝟎
−
𝒚 + 𝟐 𝟐
𝟏𝟎
= 𝟏
11. (𝒙 − 𝟏) 𝟐
𝟐𝟎
−
𝒚 + 𝟐 𝟐
𝟏𝟎
= 𝟏
Center (h, k) = (1, -2)
Vertices (h±a, k) 1 + 2 5, −2 , 1 − 2 5, −2
Foci (h±c, k) 1 + 30, −2 , 1 − 30, −2
Equation of the Transverse Axis: y=k, y = -2
Length of the Transverse Axis: 2a2(2 5)= 4 5
Equation of the Conjugate Axis: x=h, x = 1
Length of the Conjugate Axis: 2b, 2( 10)= 2 10
12. Asymptotes: 𝑦 = 𝑘 ±
𝑏
𝑎
𝑥 − ℎ
• 𝑦 = −2 ±
10
2 5
(𝑥 − 1)
• 𝑦 = −2 ±
1
2
10
5
(𝑥 − 1)
• 𝒚 = −𝟐 ±
𝟐
𝟐
(𝒙 − 𝟏)
Since x2, therefore the value of a is found under x2
𝑎 = 20 = 2 5 𝑏 = 10 𝑐 = 𝑎2 + 𝑏2 = 20 + 10 = 30
13. Write the equation of the hyperbola in standard and general
form that satisfies the given conditions. The center is at (7, -2), a
vertex is at (2, -2) and an endpoint of a conjugate axis is at (7, -6).
14. Write the equation of the hyperbola in standard and general
form that satisfies the given conditions. The center is at (7, -2), a
vertex is at (2, -2) and an endpoint of a conjugate axis is at (7, -6).
Solution: In order to write the equation, you only need the value for the
center, a and b. Then determine where the hyperbola is facing and the
appropriate standard form to be used.
15. The center (h, k) = (7, -2) is midway between the vertices.
The distance between the vertex and the center is |7-5|=5, hence a = 5.
The distance between the endpoint of the conjugate axis and the center is|-6 – (-2)|= 4, hence, b= 4. If a = 5 and b
= 4, then, 𝑐 = 𝑎2 + 𝑏2 = 52 + 42 = 41
The center and a vertex are both contained on the focal axis. Since they have the same y -coordinate, the focal axis
is horizontal. Therefor the, equation of the ellipse is of the form
(𝑥−ℎ)2
𝑎2 −
𝑦−𝑘 2
𝑏2 = 1
16. By substitution, the equation of the ellipse in
standard form is
(𝒙−𝟕) 𝟐
𝟐𝟓
−
𝒚+𝟐 𝟐
𝟏𝟔
= 𝟏.
17. Solution: To write it in general form
follow the following steps:
Step 1: Multiply both sides by the product of 25 and 16
𝟐𝟓(𝟏𝟔)
(𝒙−𝟕) 𝟐
𝟐𝟓
−
𝒚+𝟐 𝟐
𝟏𝟔
= 𝟏 𝟐𝟓(𝟏𝟔)
21. 𝟏𝟔𝒙 𝟐
− 𝟐𝟐𝟒𝒙 + 𝟕𝟖𝟒 − 𝟐𝟓𝒚 𝟐
− 𝟏𝟎𝟎𝒚 − 𝟏𝟎𝟎 − 𝟒𝟎𝟎 = 𝟎
Step 5:Combine the constant then arrange it in order
𝟏𝟔𝒙 𝟐
− 𝟐𝟓𝒚 𝟐
− 𝟐𝟐𝟒𝒙 − 𝟏𝟎𝟎𝒚 + 𝟐𝟖𝟒 = 𝟎
The standard form is
(𝒙−𝟕) 𝟐
𝟐𝟓
−
𝒚+𝟐 𝟐
𝟏𝟔
= 𝟏 and the general equation form is
𝟏𝟔𝒙 𝟐 − 𝟐𝟓𝒚 𝟐 − 𝟐𝟐𝟒𝒙 − 𝟏𝟎𝟎𝒚 + 𝟐𝟖𝟒 = 𝟎
22. An architect designs two houses that are shaped and
positioned like a part of the branches of the hyperbola
whose equation is 625𝑦2
− 400𝑥2
= 250,000where x and
y are in yards. How far apart are the houses at their closest
point?
23. An architect designs two houses that are shaped and
positioned like a part of the branches of the hyperbola
whose equation is 625𝑦2
− 400𝑥2
= 250,000where x and
y are in yards. How far apart are the houses at their closest
point?
24. Solution: The closest point of these houses are the vertices of
the hyperbola. In order to determine their distances, we need
to write the equation in standard form.