The document discusses the properties of a rectangular hyperbola. A rectangular hyperbola is a hyperbola whose asymptotes are perpendicular to each other. The equation of a rectangular hyperbola in standard form is x2/a2 - y2/a2 = 1, where the foci are located at (�a, 0) and the distance between the directrices is 2a. Rotating the hyperbola 45 degrees anticlockwise transforms it into the form x-y=�a2, with the foci now located at (a, a).
1. The document lists various trigonometric formulae including definitions of radians, trigonometric ratios, domains and ranges, allied angle relations, sum and difference formulae, and solutions to trigonometric equations.
2. Key formulae include the definitions of radians as 180°/π and degrees as π/180 radians, trigonometric ratios in terms of sine and cosine, and multiple angle formulae for sine, cosine, and tangent of doubled angles.
3. Trigonometric functions are also defined over their domains, with ranges between -1 and 1 except for cosecant, secant, and cotangent. Basic trigonometric identities and relations between quadrants are also provided.
A hyperbola is the set of all points where the absolute difference between the distance to two fixed points (foci) is a constant. It is formed by the intersection of a plane with a double cone. The key parts of a hyperbola include the foci, vertices, center, and asymptotes. Hyperbolas can be written in standard form equations and graphed based on identifying these key parts.
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.
1. The document lists various trigonometric formulae including definitions of radians, trigonometric ratios, domains and ranges, allied angle relations, sum and difference formulae, and solutions to trigonometric equations.
2. Key formulae include the definitions of radians as 180°/π and degrees as π/180 radians, trigonometric ratios in terms of sine and cosine, and multiple angle formulae for sine, cosine, and tangent of doubled angles.
3. Trigonometric functions are also defined over their domains, with ranges between -1 and 1 except for cosecant, secant, and cotangent. Basic trigonometric identities and relations between quadrants are also provided.
A hyperbola is the set of all points where the absolute difference between the distance to two fixed points (foci) is a constant. It is formed by the intersection of a plane with a double cone. The key parts of a hyperbola include the foci, vertices, center, and asymptotes. Hyperbolas can be written in standard form equations and graphed based on identifying these key parts.
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.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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.
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.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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.
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
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
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.
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
5. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
b b
1
a a
b2 a 2
ba
hyperbola has the equation;
x2 y2
2
2 1
a a
x2 y2 a2
6. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
b b
1
a a
b2 a 2
ba
hyperbola has the equation; a 2 e 2 1 a 2
x2 y2
2
2 1
a a
x2 y2 a2
7. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
b b
1
a a
b2 a 2
ba
hyperbola has the equation; a 2 e 2 1 a 2
x2 y2 e2 1 1
2
2 1
a a
e2 2
x2 y2 a2
e 2
8. Rectangular Hyperbola
A hyperbola whose asymptotes are perpendicular to each other
b b
1
a a
b2 a 2
ba
hyperbola has the equation; a 2 e 2 1 a 2
x2 y2 e2 1 1
2
2 1
a a
e2 2
x2 y2 a2
e 2
eccentricity is 2
10. y Y
In order to make the
P x, y asymptotes the coordinate
axes we need to rotate the
x curve 45 degrees
anticlockwise.
X
11. y Y
In order to make the
P x, y asymptotes the coordinate
axes we need to rotate the
x curve 45 degrees
anticlockwise.
X
i.e. P x, y x iy is multiplied by cis 45
12. y Y
In order to make the
P x, y asymptotes the coordinate
axes we need to rotate the
x curve 45 degrees
anticlockwise.
X
i.e. P x, y x iy is multiplied by cis 45
x iy cos 45 i sin 45
x iy
1 1
i
2 2
13. y Y
In order to make the
P x, y asymptotes the coordinate
axes we need to rotate the
x curve 45 degrees
anticlockwise.
X
i.e. P x, y x iy is multiplied by cis 45
x iy cos 45 i sin 45
x iy
1 1
i
2 2
1
x iy 1 i
2
1
x ix iy y
2
x y x y
i
2 2
14. y Y
In order to make the
P x, y asymptotes the coordinate
axes we need to rotate the
x curve 45 degrees
anticlockwise.
X
i.e. P x, y x iy is multiplied by cis 45
x iy cos 45 i sin 45
1 i 1 x y x y
x iy X Y
2 2 2 2
1
x iy 1 i
2
1
x ix iy y
2
x y x y
i
2 2
15. y Y
In order to make the
P x, y asymptotes the coordinate
axes we need to rotate the
x curve 45 degrees
anticlockwise.
X
i.e. P x, y x iy is multiplied by cis 45
x iy cos 45 i sin 45
1 i 1 x y x y
x iy X Y
2 2 2 2
1 x2 y2
x iy 1 i XY
2 2
1
x ix iy y
2
x y x y
i
2 2
16. y Y
In order to make the
P x, y asymptotes the coordinate
axes we need to rotate the
x curve 45 degrees
anticlockwise.
X
i.e. P x, y x iy is multiplied by cis 45
x iy cos 45 i sin 45
1 i 1 x y x y
x iy X Y
2 2 2 2
1 x2 y2
x iy 1 i XY
2 2
1 a2
x ix iy y XY
2 2
x y x y
i
2 2
19. focus; ae,0
2a,0
1 1 i
2a
2 2
a ai
focus a, a
20. a
focus; ae,0 directrix; x
e
2a,0 a
x
2
1 1 i
2a
2 2
a ai
focus a, a
21. a
focus; ae,0 directrix; x
e
2a,0 a
x
2
1 1 i
2a directrices are || to y axis
2 2
when rotated || to y x
a ai
focus a, a
22. a
focus; ae,0 directrix; x
e
2a,0 a
x
2
1 1 i
2a directrices are || to y axis
2 2
when rotated || to y x
a ai
thus in form x y k 0
focus a, a
23. a
focus; ae,0 directrix; x
e
2a,0 a
x
2
1 1 i
2a directrices are || to y axis
2 2
when rotated || to y x
a ai
thus in form x y k 0
focus a, a 2a
Now distance between directrices is
2
24. a
focus; ae,0 directrix; x
e
2a,0 a
x
2
1 1 i
2a directrices are || to y axis
2 2
when rotated || to y x
a ai
thus in form x y k 0
focus a, a 2a
Now distance between directrices is
2
a
distance from origin to directrix is
2
25. a
focus; ae,0 directrix; x
e
2a,0 a
x
2
1 1 i
2a directrices are || to y axis
2 2
when rotated || to y x
a ai
thus in form x y k 0
focus a, a 2a
Now distance between directrices is
2
a
distance from origin to directrix is
2
00k a
2 2
26. a
focus; ae,0 directrix; x
e
2a,0 a
x
2
1 1 i
2a directrices are || to y axis
2 2
when rotated || to y x
a ai
thus in form x y k 0
focus a, a 2a
Now distance between directrices is
2
a
distance from origin to directrix is
2
00k a
2 2
k a
k a
27. a
focus; ae,0 directrix; x
e
2a,0 a
x
2
1 1 i
2a directrices are || to y axis
2 2
when rotated || to y x
a ai
thus in form x y k 0
focus a, a 2a
Now distance between directrices is
2
a
distance from origin to directrix is
2
00k a
2 2
k a
k a
directrices are x y a
28. The rectangular hyperbola with x and y axes as aymptotes,
has the equation;
1 2
xy a
2
where;
foci : a, a
directrices : x y a
eccentricity 2
29. The rectangular hyperbola with x and y axes as aymptotes,
has the equation;
1 2
xy a
2
where;
foci : a, a
directrices : x y a
eccentricity 2
Parametric Coordinates of xy c 2
30. The rectangular hyperbola with x and y axes as aymptotes,
has the equation;
1 2
xy a
2
where;
foci : a, a
directrices : x y a
eccentricity 2
Parametric Coordinates of xy c 2
c
x ct y
t
31. The rectangular hyperbola with x and y axes as aymptotes,
has the equation;
1 2
xy a
2
where;
foci : a, a
directrices : x y a
eccentricity 2
Parametric Coordinates of xy c 2
c
x ct y
t
Tangent: x t 2 y 2ct
32. The rectangular hyperbola with x and y axes as aymptotes,
has the equation;
1 2
xy a
2
where;
foci : a, a
directrices : x y a
eccentricity 2
Parametric Coordinates of xy c 2
c
x ct y
t
Tangent: x t 2 y 2ct Normal: t 3 x ty ct 4 1
33. e.g. (i) (1991)
The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
34. e.g. (i) (1991)
The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
y y=x
x
35. e.g. (i) (1991)
The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
y y=x
xy 4
2,2
x2 4
x
2,2 x 2
36. e.g. (i) (1991)
The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
y y=x
xy 4
2,2
x2 4
x
2,2 x 2
2t , 2 is x t 2 y 4t
b) Show that the tangent at P
t
37. e.g. (i) (1991)
The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
y y=x
xy 4
2,2
x2 4
x
2,2 x 2
2t , 2 is x t 2 y 4t
b) Show that the tangent at P
t
4
y
x
dy 4
2
dx x
38. e.g. (i) (1991)
The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
y y=x
xy 4
2,2
x2 4
x
2,2 x 2
2t , 2 is x t 2 y 4t
b) Show that the tangent at P
t
4 dy 4
y when x 2t ,
x dx 2t 2
dy 4
2 1
2
dx x t
39. e.g. (i) (1991)
The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
y y=x
xy 4
2,2
x2 4
x
2,2 x 2
2t , 2 is x t 2 y 4t
b) Show that the tangent at P
t
4 dy 4 2 1
y when x 2t , y 2 x 2t
x dx 2t 2
t t
dy 4
2 1
2
dx x t
40. e.g. (i) (1991)
The hyperbola H is xy= 4
a) Sketch H showing where H intersects the axis of symmetry.
y y=x
xy 4
2,2
x2 4
x
2,2 x 2
2t , 2 is x t 2 y 4t
b) Show that the tangent at P
t
4 dy 4 2 1
y when x 2t , y 2 x 2t
x dx 2t 2
t t
dy 4 1 t 2 y 2t x 2t
2 2
dx x t x t 2 y 4t
41. s, 2
c) s 0, s t , show that the tangents at P and Q 2
2 2
s
intersect at M 4 st , 4
st st
42. s, 2
c) s 0, s t , show that the tangents at P and Q 2
2 2
s
intersect at M 4 st , 4
st st
P : x t 2 y 4t
Q : x s 2 y 4s
43. s, 2
c) s 0, s t , show that the tangents at P and Q 2
2 2
s
intersect at M 4 st , 4
st st
P : x t 2 y 4t
Q : x s 2 y 4s
t 2
s 2 y 4t 4 s
44. s, 2
c) s 0, s t , show that the tangents at P and Q 2
2 2
s
intersect at M 4 st , 4
st st
P : x t 2 y 4t
Q : x s 2 y 4s
t 2
s 2 y 4t 4 s
t s t s y 4t s
4
y
st
45. s, 2
c) s 0, s t , show that the tangents at P and Q 2
2 2
s
intersect at M 4 st , 4
st st
P : x t 2 y 4t 4t 2
x 4t
Q : x s 2 y 4s st
t 2
s 2 y 4t 4 s
t s t s y 4t s
4
y
st
46. s, 2
c) s 0, s t , show that the tangents at P and Q 2
2 2
s
intersect at M 4 st , 4
st st
P : x t 2 y 4t 4t 2
x 4t
Q : x s 2 y 4s st
t 2
s y 4t 4 s
2
x
4 st 4t 2 4t 2
st
t s t s y 4t s 4 st
4
y st
st
47. 2 s, 2
c) s 0, s t , show that the tangents at P and Q
2 2
s
intersect at M 4 st , 4
st st
P : x t 2 y 4t 4t 2
x 4t
Q : x s 2 y 4s st
t 2
s y 4t 4 s
2
x
4 st 4t 2 4t 2
st
t s t s y 4t s 4 st
4
y st
st
4 st , 4
M is
st st
48. 1
d) Suppose that s , show that the locus of M is a straight
t
line through the origin, but not including the origin.
49. 1
d) Suppose that s , show that the locus of M is a straight
t
line through the origin, but not including the origin.
4 st 4
x y
st st
50. 1
d) Suppose that s , show that the locus of M is a straight
t
line through the origin, but not including the origin.
4 st 4
x y
st st
1
s
t
st 1
51. 1
d) Suppose that s , show that the locus of M is a straight
t
line through the origin, but not including the origin.
4 st 4
x y
st st
1
s
t
st 1
4
x
st
52. 1
d) Suppose that s , show that the locus of M is a straight
t
line through the origin, but not including the origin.
4 st 4
x y
st st
1
s
t
st 1
4 y
4
x
st st
x
53. 1
d) Suppose that s , show that the locus of M is a straight
t
line through the origin, but not including the origin.
4 st 4
x y
st st
1
s
t
st 1
4 y
4
x
st st
x
y x
54. 1
d) Suppose that s , show that the locus of M is a straight
t
line through the origin, but not including the origin.
4 st 4
x y
st st
1
s
t
st 1
4 y
4
x
st st
x
4
y x 0, thus M 0,0
st
55. 1
d) Suppose that s , show that the locus of M is a straight
t
line through the origin, but not including the origin.
4 st 4
x y
st st
1
s
t
st 1
4 y
4
x
st st
x
4
y x 0, thus M 0,0
st
locus of M is y x, excluding 0,0
57. (ii) Show that PS PS 2a
y
P x, y
S S x
By definition of an ellipse;
58. (ii) Show that PS PS 2a
y
P x, y
M
S S x
a
x x
a
e e
By definition of an ellipse;
PS PS ePM
59. (ii) Show that PS PS 2a
y
P x, y
M M
S S x
a
x x
a
e e
By definition of an ellipse;
PS PS ePM ePM
60. (ii) Show that PS PS 2a
y
P x, y
M M
S S x
a
x x
a
e e
By definition of an ellipse;
PS PS ePM ePM
e PM PM
2a
e
e
2a
61. (ii) Show that PS PS 2a
y
P x, y
M M
S S x
a
x x
a
e e
By definition of an ellipse;
PS PS ePM ePM
e PM PM Exercise 6D; 3, 4, 7, 10, 11a,
2a
e
12, 14, 19, 21, 26, 29,
e 31, 43, 47
2a