Topics
Angle and Its Measurement
- Directed Angle
- Angles of Different Measurements
- Angles in Standard Position
- Measures of Angles with Various Systems
- Area of a Sector
- Length of an Arc
Trigonometry - 1
- Trigonometric Ratios
- Trigonometric Functions with the Help of a Circle
- Signs of Trigonometric Functions in Different Quadrants
- Range of Cosθ and Sinθ
- Trigonometric Functions of Specific Angles
- Trigonometric Functions of Negative Angles
- Important Identities and Standard Results
- Periodicity of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Graphs of Trigonometric Functions
- Polar Co-ordinate System
Trigonometry - 2
- Trigonometric Functions of Sum and Difference of Three Angles
- Trigonometric Functions of Allied Angels
- Trigonometric Functions of Multiple Angles
- Conversion Formulae in Trigonometry
- Trigonometric Functions of Angles of a Triangle
Determinants and Matrices
- Definition and Expansion of Determinants
- Minors and Cofactors of Elements of Determinants
- Properties of Determinants
- Application of Determinants
- Determinant Method (Cramer’s Rule)
- Consistency of Three Equations in Two Variables
- Area of Triangle and Collinearity of Three Points
- Concept of Matrices
- Types of Matrices
- Operation on Matrices
- Properties of Matrix Multiplication
- Transpose of a Matrix
Straight Line
- Locus of a Points in a Co-ordinate Plane
- Equations of Line in Different Forms
- Family & Concurrent Lines
Circle
- Equation of a Circle in some special cases
- Equation of a Circle in Different Forms
- Secant and Tangent
- Equation of Tangent and Condition of Tangency
- Tangent and Secant Properties
- Director circle
Conic Sections
- Double Cone
- Fundamentals of Conic Sections
- Parabola and its types
- Ellipse and its Types
- Hyperbola and its Types
Measures of Dispersion
- Meaning and Definition of Dispersion
- Measures of Dispersion
- Quartiles and Range in Statistics
- Variance
- Standard Deviation
- Change of Origin and Scale of Variance and Standard Deviation
- Standard Deviation for Combined Data
- Coefficient of Variation
Probability
- Basic Terminologies
- Elementary Types of Events and Properties of Probability
- Concept of Probability
- Addition Theorem for Two Events
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Odds (Ratio of Two Complementary Probabilities)
Complex Numbers
- Introduction of Complex Number
- Concept of Complex Numbers
- Algebraic Operations of Complex Numbers
- Square Root of a Complex Number
- Fundamental Theorem of Algebra
- Argand's Diagram
- De Moivres Theorem
- Cube Root of Unity
- Set of Points in Complex Plane
Sequences and Series
- Sequence, Series, and Progression
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- Harmonic Progression (H. P.)
- Arithmetico Geometric Series
- Power Series
Permutations and Combination
- Fundamental Principles of Counting
- Invariance Principle
- Factorial Notation
- Permutations
- Circular Permutations
- Combination
Methods of Induction and Binomial Theorem
- Principle of Mathematical Induction
- Binomial Theorem for Positive Integral Index
- General Term in Expansion of (a + b)n
- Middle term(s) in the expansion of (a + b)n
- Binomial Theorem for Negative Index Or Fraction
- Binomial Coefficients
Sets and Relations
- Sets and Their Representations
- Classification of Sets
- Fundamental Concepts of Ordered Pairs and Relations
- Intervals
Functions
- Domain and Range of a Function
- Algebra of Functions
Limits
- Concept of Limits
- Factorization Method
- Rationalization Method
- Limits of Trigonometric Functions
- Substitution Method
- Limits of Exponential and Logarithmic Functions
- Limit at Infinity
Continuity
- Continuous and Discontinuous Functions
Differentiation
- Definition of Derivative and Differentiability
- Rules of Differentiation (Without Proof)
- Derivative of Algebraic Functions
- Derivatives of Trigonometric Functions
- Derivative of Logarithmic Functions
- Derivatives of Exponential Functions
- L' Hospital'S Theorem
Notes
The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis or y-axis. The four possible such orientations of parabola are shown in following fig.
We will derive the equation for the parabola shown above in First Fig with focus at (a, 0) a > 0; and directricx x = – a as below:
Let F be the focus and l the directrix. Let FM be perpendicular to the directrix and bisect FM at the point O. Produce MO to X. By the definition of parabola, the mid-point O is on the parabola and is called the vertex of the parabola. Take O as origin, OX the x-axis and OY perpendicular to it as the y-axis. Let the distance from the directrix to the focus be 2a. Then, the coordinates of the focus are (a, 0), and the equation of the directrix is x + a = 0 as in above fig.
Let P(x, y) be any point on the parabola such that
PF = PB, ... (1)
where PB is perpendicular to l. The coordinates of B are (– a, y).
By the distance formula, we have
PF = `sqrt((x-a)^2+y^2) `and PB = `sqrt(x + a)^2`
Since PF = PB, we have
`sqrt((x-a)^2 + y^2) = sqrt((x+a)^2)`
i.e. `(x – a)^2 + y^2 = (x + a)^2
or x^2 – 2ax + a^2 + y^2 = x^2 + 2ax + a^2`
or `y^2` = 4ax ( a > 0).
Hence, any point on the parabola satisfies
`y^2` = 4ax. ...(2)
Conversely, let P(x, y) satisfy the equation (2)
PF = `sqrt((x-a)^2+y^2) = sqrt((x-a)^2 + 4ax)`
= `sqrt(x+a)^2` = PB .....(3)
and so P(x,y) lies on the parabola.
Standard equation of Parabola:
`y^2` = 4ax
`y^2` = - 4ax
`x^2` = 4ay
`x^2` = - 4ay
Definition
A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane.
Notes
The fixed line is called the directrix of the parabola and the fixed point F is called the focus Fig. (‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when you throw a ball in the air).
A line through the focus and perpendicular to the directrix is called the axis of the parabola. The point of intersection of parabola with the axis is called the vertex of the parabola Fig.
