The document discusses conic sections, specifically the ellipse. An ellipse is defined as the locus of points where the sum of the distances from two fixed points (foci) is a constant. For an ellipse, the eccentricity e is less than 1. The foci of an ellipse lie on its major axis, and the distance from each focus to any point on the ellipse is e times the distance from that point to a line (directrix). The equation of an ellipse presented is x^2/a^2 + y^2/b^2 = 1, where a and b are the semi-major and semi-minor axes and b^2 = a^2(1 - e^2).
The document discusses conic sections, which are curves defined by a fixed point (focus) and fixed line (directrix) where the distance from any point on the curve to the focus is a constant multiple of its distance to the directrix. Circles have an eccentricity (e) of 0, ellipses have e < 1, parabolas have e = 1, and hyperbolas have e > 1. Ellipses are defined such that the sum of the distances from any point to the two foci is a constant. Key properties of ellipses include their foci, located at (±ae, 0), their directrices defined by x = ±a/e, and the relationship between the semi-major
11X1 T10 05 sum of an arithmetic seriesNigel Simmons
The document discusses methods for calculating the sum of terms in an arithmetic series. It provides the general formula for the sum as Sn = (a + l)n/2 where a is the first term, l is the last term, n is the number of terms, and d is the common difference. It also shows how to calculate the sum given specific values of a, d, and n. Examples are worked through demonstrating how to use the formulas.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document discusses conic sections, which are curves defined by a fixed point (focus) and fixed line (directrix) where the distance from any point on the curve to the focus is a constant multiple of its distance to the directrix. Circles have an eccentricity (e) of 0, ellipses have e < 1, parabolas have e = 1, and hyperbolas have e > 1. Ellipses are defined such that the sum of the distances from any point to the two foci is a constant. Key properties of ellipses include their foci, located at (±ae, 0), their directrices defined by x = ±a/e, and the relationship between the semi-major
11X1 T10 05 sum of an arithmetic seriesNigel Simmons
The document discusses methods for calculating the sum of terms in an arithmetic series. It provides the general formula for the sum as Sn = (a + l)n/2 where a is the first term, l is the last term, n is the number of terms, and d is the common difference. It also shows how to calculate the sum given specific values of a, d, and n. Examples are worked through demonstrating how to use the formulas.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
2. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
3. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
4. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
e=0 circle
5. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
e=0 circle
e<1 ellipse
6. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
e=0 circle
e<1 ellipse
e=1 parabola
7. Conics
The locus of points whose distance from a fixed point (focus) is a
multiple, e, (eccentricity) of its distance from a fixed line (directrix)
e=0 circle
e<1 ellipse
e=1 parabola
e>1 hyperbola
10. Ellipse (e < 1)
y
b
A’ A
-a S a Z x
-b
SA = eAZ and SA’ = eA’Z
11. Ellipse (e < 1)
y
b
A’ A
-a S a Z x
-b
SA = eAZ and SA’ = eA’Z
(1) SA’ + SA = 2a
(2) SA’ – SA = e(A’Z – AZ)
12. Ellipse (e < 1)
y
b
A’ A
-a S a Z x
-b
SA = eAZ and SA’ = eA’Z
(1) SA’ + SA = 2a
(2) SA’ – SA = e(A’Z – AZ)
= e(AA’)
= e(2a)
= 2ae
13. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e)
SA’ = a(1 + e)
14. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
15. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus
OS = OA - SA
16. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus
OS = OA - SA
= a – a(1 – e)
= ae
S ae,0
17. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus Directrix
OS = OA - SA OZ = OA + AZ
= a – a(1 – e)
= ae
S ae,0
18. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus Directrix
OS = OA - SA OZ = OA + AZ
SA
= a – a(1 – e) OA SA eAZ
= ae e
S ae,0
19. b
A’ A
-a S a Z x
-b
(1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e)
SA’ = a(1 + e) SA = a(1 - e)
Focus Directrix
OS = OA - SA OZ = OA + AZ
SA
= a – a(1 – e) OA SA eAZ
= ae e
ae a1 e
S ae,0
e e a
a directrices x
e
e
20. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
21. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
SP ePN
22. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
SP ePN
2
x ae 2 y 02 e x y y 2
a
e
2
2 a
x ae y e x
2 2
e
23. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
SP ePN
2
x ae 2 y 02 e x y y 2
a
e
2
2 a
x ae y e x
2 2
e
x 2 2aex a 2 e 2 y 2 e 2 x 2 2aex a 2
x 2 1 e 2 y 2 a 2 1 e 2
24. S ae,0
b P
P x, y
N
A’ A
-a a
N , y
a S Z x
e -b
SP ePN
2
x ae 2 y 02 e x y y 2
a
e
2
2 a
x ae y e x
2 2
e
x 2 2aex a 2 e 2 y 2 e 2 x 2 2aex a 2
x 2 1 e 2 y 2 a 2 1 e 2
x2 y2
2 1
a a 1 e
2 2
25. b2
when x 0, y b 1
a 1 e
i.e. 2 2
b 2 a 2 1 e 2
26. b2
when x 0, y b 1
a 1 e
i.e. 2 2
b 2 a 2 1 e 2
Ellipse: (a > b) x2 y2
2
2 1
a b
where; b 2 a 2 1 e 2
focus : ae,0
a
directrices : x
e
e is the eccentricity
major semi-axis = a units
minor semi-axis = b units
27. b2
when x 0, y b 1
a 1 e
i.e. 2 2
b 2 a 2 1 e 2
Ellipse: (a > b) x2 y2 Note: If b > a
2
2 1
a b foci on the y axis
where; b a 1 e
2 2 2
a 2 b 2 1 e 2
focus : ae,0 focus : 0,be
a b
directrices : x directrices : y
e e
e is the eccentricity
major semi-axis = a units
minor semi-axis = b units
28. b2
when x 0, y b 1
a 1 e
i.e. 2 2
b 2 a 2 1 e 2
Ellipse: (a > b) x2 y2 Note: If b > a
2
2 1
a b foci on the y axis
where; b a 1 e
2 2 2
a 2 b 2 1 e 2
focus : ae,0 focus : 0,be
a b
directrices : x directrices : y
e e
e is the eccentricity
major semi-axis = a units Area ab
minor semi-axis = b units
29. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
30. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
x2 y2
1
9 5
a2 9
a3
31. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
x2 y2
1 b2 5
9 5
a 2 1 e 2 5
a2 9
a3
32. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
x2 y2
1 b2 5
9 5
a 2 1 e 2 5
91 e 2 5
a2 9
a3
5
1 e 2
9
4
e
2
9
2
e
3
33. e.g. Find the eccentricity, foci and directrices of the ellipse
x2 y2
1 and sketch the ellipse showing all of the important
9 5
features.
x2 y2
1 b2 5
9 5
a 2 1 e 2 5
91 e 2 5
a2 9
2
a3 eccentricity
5 3
1 e 2
9 foci : 2,0
4
e
2
3
9 directrices : x 3
2
2
e 9
3 x
2
47. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
48. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
49. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
50. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
centre : (1,2)
51. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
centre : (1,2)
b2 9
b3
52. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
centre : (1,2)
b2 9 a 2 b 2 1 e 2
b3 4 91 e 2
5
e
3
53. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22
1
4 9
centre : (1,2)
b2 9 a 2 b 2 1 e 2
b3 4 91 e 2
5
e
3
foci : 1,2 5
9
directrices : y 2
5
54. (ii) 9 x 2 4 y 2 18 x 16 y 11 0
x 2 2 x y 2 4 y 11
4 9 36
x 12 y 22 11 1 4
4 9 36 4 9
x 12 y 22 Exercise 6A; 1, 2, 3, 5, 7,
1
4 9 8, 9, 11, 13, 15
centre : (1,2)
b2 9 a 2 b 2 1 e 2
b3 4 91 e 2
5
e
3
foci : 1,2 5
9
directrices : y 2
5