Topics
Mathematical Logic
Matrices
Differentiation
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Inverse Functions
- Derivatives of Logarithmic Functions
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Second Order Derivative
- Overview of Differentiation
Applications of Derivatives
Integration
Definite Integration
Applications of Definite Integration
- Standard Forms of Parabola and Their Shapes
- Ellipse and its Types
- Area Under Simple Curves
- Overview of Application of Definite Integration
Differential Equation and Applications
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equation by Eliminating Arbitary Constant
- Differential Equations with Variables Separable Method
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Overview of Differential Equations
Commission, Brokerage and Discount
- Commission and Brokerage Agent
- Concept of Discount
- Overview of Commission, Brokerage and Discount
Insurance and Annuity
- Insurance
- Types of Insurance
- Annuity
- Overview of Insurance and Annuity
Linear Regression
- Regression
- Types of Linear Regression
- Fitting Simple Linear Regression
- The Method of Least Squares
- Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
- Properties of Regression Coefficients
- Overview: Linear Regression
Time Series
- Introduction to Time Series
- Uses of Time Series Analysis
- Components of a Time Series
- Mathematical Models
- Measurement of Secular Trend
- Overview of Time Series
Index Numbers
- Weighted Aggregate Method
- Cost of Living Index Number
- Method of Constructing Cost of Living Index Numbers - Aggregative Expenditure Method
- Overview of Index Numbers
- Method of Constructing Cost of Living Index Numbers - Family Budget Method
- Uses of Cost of Living Index Number
Linear Programming
- Introduction of Linear Programming
- Linear Programming Problem (L.P.P.)
- Mathematical Formulation of Linear Programming Problem
- Overview of Linear Programming
Assignment Problem and Sequencing
- Assignment Problem
- Hungarian Method of Solving Assignment Problem
- Special Cases of Assignment Problem
- Sequencing Problem
- Types of Sequencing Problem
- Finding an Optimal Sequence
- Overview of Assignment Problem and Sequencing
Probability Distributions
- Poisson Distribution
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
- Overview of Binomial Distribution
- Standard equation of the ellipse
- Special cases of an ellipse
- Tangent to an ellipse
- Equation of tangent to the ellipse
- Condition for tangency
- Tangents from a point to the ellipse
- Locus of point of intersection of perpendicular tangents
- Auxilary circle and director circle of the ellipse
Definition
An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
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.
Notes
The two fixed points are called the foci (plural of ‘focus’) of the ellipse in the following Fig.
The mid point of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called the minor axis. The end points of the major axis are called the vertices of the ellipse in following fig.
