MATHEMATICS

GRADE 4TH TO BACHELOR CLASSES (BA/B.Sc./B.E./B.TECH.) and other Engineering Courses

MATHEMATICS – 10, 20, 20-1, 30, 30-1, 31, 100, 101, 102, 113, 114, 115, 120 AND 125

Algebra - Linear, Radical, Quadratic, Monomial/Binomial/Polynomial, Exponential, Logarithmic, Rational, and Trigonometric Equations, Algebraic and Graphic expressions, relations, and functions and more

Trigonometry - Trigonometry equations, Trigonometric functions, and Trigonometric ratios etc.

Calculus - Differential Calculus, Integral Calculus, Derivatives, Differential Equations, Integral Equations and more

Determinants and Matrices, Vectors and Vector space, Rank, Eigenvalues and Eigenvectors, and more

Straight Line and Geometry - Point formula, distance formula, mid-point formula, section formula, Angles and Area of Triangles, Two Dimensional and Three Dimensional Geometry and more

Co-ordinates and Solid Geometry - Coordinates, Planes, Coordinate Systems : One Dimensional, Two Dimensional and Three Dimensional, Area and Perimeter of Polygons, Transformation and Rotations, Equations of Curves, circles, and Ellipses, Parabola, Hyperbola, and more

Statistics - Mean, Median, Mode, Co-relation, rank, rank of co-relation, Deviations, mean deviation, standard deviation and more

An ellipse can be defined as the locus of all points that satisfy the equation x^2 / a^2 + y^2 / b^2 = 1

where:

x,y are the coordinates of any point on the ellipse,

a, b are the radius on the x and y axes respectively

Parametric Equation of an Ellipse

An ellipse can be defined as the locus of all points that satisfy the equations

x = a cos t

y = b sin t

where:

x,y are the coordinates of any point on the ellipse,

a, b are the radius on the x and y axes respectively,

t is the parameter, which ranges from 0 to 2π radians.

This equation is very similar to the one used to define a circle, and much of the discussion is omitted here to avoid duplication.

The only difference between the circle and the ellipse is that in a circle there is one radius, but an ellipse has two:

One radius is measured along the x-axis and is usually called a.
The other is measured along the y-axis and is usually called b.

x^2 + y^2 = 0, Circle x^2 + y^2 = r^2.

Ellipse x^2 / a^2 + y^2 / b^2 = 1, Ellipse x^2 / b^2 + y^2 / a^2 = 1,

Hyperbola x^2 / a^2 - y^2 / b^2 = 1. Parabola 4px = y^2 ...

The most general form of a quadratic function

The graphs of quadratic functions are called parabolas. Here are some examples of parabolas.

All parabolas are vaguely “U” shaped and they will have a highest or lowest point that is called the vertex. Parabolas may open up or down and may or may not have x-intercepts and they will always have a single y-intercept.

Note as well that a parabola that opens down will always open down and a parabola that opens up will always open up. In other words a parabola will not all of a sudden turn around and start opening up, if it has already started opening down. Similarly, if it has already started opening up it will not turn around and start opening down all of a sudden.

The dashed line with each of these parabolas is called the axis of symmetry. Every parabola has an axis of symmetry and, as the graph shows, the graph to either side of the axis of symmetry is a mirror image of the other side. This means that if we know a point on one side of the parabola we will also know a point on the other side based on the axis of symmetry. We will see how to find this point once we get into some examples.

We should probably do a quick review of intercepts before going much farther. Intercepts are the points where the graph will cross the x or y-axis. We also saw a graph in the section where we introduced intercepts where an intercept just touched the axis without actually crossing it.

Finding intercepts is a fairly simple process. To find the y-intercept of a function all we need to do is set and evaluate to find the y coordinate. In other words, the y-intercept is the point . We find x-intercepts in pretty much the same way. We set and solve the resulting equation for the x coordinates. So, we will need to solve the equation.

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