Topics
Trigonometric Functions
- Concept of Angle
- Introduction of Trigonometric Functions
- Signs of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Trigonometric Functions of Sum and Difference of Two Angles
- Trigonometric Equations
- Truth of the Identity
- Negative Function Or Trigonometric Functions of Negative Angles
- 90 Degree Plusminus X Function
- Conversion from One Measure to Another
- 180 Degree Plusminus X Function
- 2X Function
- 3X Function
- Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications
- Graphs of Trigonometric Functions
- Transformation Formulae
- Values of Trigonometric Functions at Multiples and Submultiples of an Angle
- Sine and Cosine Formulae and Their Applications
Binomial Theorem
Statistics
- Measures of Dispersion
- Concept of Range
- Mean Deviation
- Introduction of Variance and Standard Deviation
- Standard Deviation
- Standard Deviation of a Discrete Frequency Distribution
- Standard Deviation of a Continuous Frequency Distribution
- Shortcut Method to Find Variance and Standard Deviation
- Introduction of Analysis of Frequency Distributions
- Comparison of Two Frequency Distributions with Same Mean
- Statistics Concept
- Central Tendency - Mean
- Central Tendency - Median
- Concept of Mode
- Measures of Dispersion - Quartile Deviation
- Standard Deviation - by Short Cut Method
Sets and Functions
Limits and Derivatives
- Intuitive Idea of Derivatives
- Introduction of Limits
- Introduction to Calculus
- Algebra of Limits
- Limits of Polynomials and Rational Functions
- Limits of Trigonometric Functions
- Introduction of Derivatives
- Algebra of Derivative of Functions
- Derivative of Polynomials and Trigonometric Functions
- Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically
- Limits of Logarithmic Functions
- Limits of Exponential Functions
- Derivative of Slope of Tangent of the Curve
- Theorem for Any Positive Integer n
- Graphical Interpretation of Derivative
- Derive Derivation of x^n
Mathematical Reasoning
Straight Lines
Introduction to Three-dimensional Geometry
Probability
Algebra
Relations and Functions
- Cartesian Product of Sets
- Relation
- Concept of Functions
- Some Functions and Their Graphs
- Algebra of Real Functions
- Ordered Pairs
- Equality of Ordered Pairs
- Pictorial Diagrams
- Graph of Function
- Pictorial Representation of a Function
- Exponential Function
- Logarithmic Functions
- Brief Review of Cartesian System of Rectanglar Co-ordinates
Sequence and Series
Linear Inequalities
Coordinate Geometry
Sets
- Sets and Their Representations
- The Empty Set
- Finite and Infinite Sets
- Equal Sets
- Subsets
- Power Set
- Universal Set
- Venn Diagrams
- Intrdouction of Operations on Sets
- Union Set
- Intersection of Sets
- Difference of Sets
- Complement of a Set
- Practical Problems on Union and Intersection of Two Sets
- Proper and Improper Subset
- Open and Close Intervals
- Operation on Set - Disjoint Sets
- Element Count Set
Conic Sections
- Sections of a Cone
- Concept of Circle
- Introduction of Parabola
- Standard Equations of Parabola
- Latus Rectum
- Introduction of Ellipse
- Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus from the Centre of the Ellipse
- Special Cases of an Ellipse
- Eccentricity
- Standard Equations of an Ellipse
- Latus Rectum
- Introduction of Hyperbola
- Eccentricity
- Standard Equation of Hyperbola
- Latus Rectum
- Standard Equation of a Circle
Calculus
Complex Numbers and Quadratic Equations
Permutations and Combinations
- Fundamental Principles of Counting
- Permutations
- Combination
- Introduction of Permutations and Combinations
- Permutation Formula to Rescue and Type of Permutation
- Smaller Set from Bigger Set
- Derivation of Formulae and Their Connections
- Simple Applications of Permutations and Combinations
- Factorial N (N!) Permutations and Combinations
Mathematical Reasoning
Statistics and Probability
Principle of Mathematical Induction
notes
The equation of an ellipse is simplest if the centre of the ellipse is at the origin and the foci are
on the x-axis or y-axis. The two such possible orientations are shown in above Fig.
We will derive the equation for the ellipse shown above in Fig with foci on the x-axis.
Let `F_1` and `F_2` be the foci and O be the midpoint of the line segment
`F_1F_2`. Let O be the origin and the line from O through `F_2` be the positive x-axis and that through `F_1`as the negative x-axis. Let, the line through O perpendicular to the x-axis be the y-axis. Let the coordinates of `F_1` be
(– c, 0) and `F_2` be (c, 0).
Let P(x, y) be any point on the ellipse such that the sum of the distances from P to the two foci be 2a so given
`PF_1` + `PF_2` = 2a. ... (1)
Using the distance formula,
`sqrt((x + c)^2 + y^2) + sqrt((x - c )^2 + y^2) 2a`
i.e., `sqrt ((x + c)^2 + y^2 ) = 2a - sqrt((x - c )^2 + y^2)`
Squaring both sides, we get
`(x + c)^2 + y^2 = 4a^2 – 4a sqrt((x-c)^2 + y^2) + (x - c)^2 + y^2`
which on simplification gives
`sqrt((x - c)^2 + y^2) = a-(c/a) x`
Squaring again and simplifying, we get
`x^2/a^2 + y^2/(a^2 - c^2) = 1`
i.e., `x^2/a^2 + y^2/b^2 = 1`
Hence any point on the ellipse satisfies
`x^2/a^2 + y^2/b^2 = 1`
The standard equation of the ellipses for the above Fig.
1. Ellipse is symmetric with respect to both the coordinate axes since if (x, y) is a point on the ellipse, then (– x, y), (x, –y) and (– x, –y) are also points on the ellipse.
2. The foci always lie on the major axis. The major axis can be determined by finding the intercepts on the axes of symmetry. That is, major axis is along the x-axis if the coefficient of `x^2` has the larger denominator and it is along the y-axis if the coefficient of `y^2` has the larger denominator.
