Introduction to Three–dimensional Geometry: Coordinate axes and coordinate planes in three dimensions To locate a point in a plane, two numbers are needed. Recall that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate. For this reason, … Read more
Conic sections: Standard equation and simple properties of Hyperbola The standard form of the equation of a hyperbola is developed in a similar methodology to an ellipse. Note, however, that a, b and c are related differently for hyperbolas than for ellipses.For a hyperbola, the distance between the foci and the centre is greater than … Read more
Conic sections: Hyperbola − a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. When we pull the two foci of an ellipse so far apart that they moved outside the ellipse, the hyperbola, another conic section is formed. Although ellipses and hyperbolas have completely different … Read more
For a parabola with the axis of symmetry parallel to the y-axis and vertex at (h, k), the standard form is (x – h)2 = 4p(y – k) Example 1: Find the equation of the parabola that has a minimum at(-2, 6) and passes through the point (2, 8). The axis of symmetry is parallel … Read more
Geometric Definition of a Parabola: The collection of all the points P(x,y) in a plane at the same distance from a fixed point, the focus, as they are from a fixed line called the directrix. Note that “p” represents the distance from the focus to the vertex or the distance from the vertex to the … Read more
The equation of an ellipse with its centre at the origin has one of two forms: Horizontal Ellipse Foci: Vertical Ellipse: Foci: For a horizontal Major Axis and C(0,0): major axis = 2a and minor axis = 2b For a Horizontal Major Axis … Read more
An ellipse can informally be explained as an oval or a “squished” circle. It is defined as the set of all points in a plane such that the distance from two fixed points (foci) on the plane is constant. The major axis is the axis on which the foci lie; the longer axis of symmetry … Read more
Set of all points in a plane at a fixed distance (radius) from a fixed point (centre). Let the radius of a circle be r and the centre be (h, k) and suppose point P(x, y) be any point on the circle. This means that the distance between (x, y) and (h, k) is r.By … Read more
The geometric definition of a circle can be explained by a plane intersecting a circle. A circle is created when the set of all points that are equidistant from a given point (the centre). Radius is a distance between the centre and any point on the circumference. The diameter cuts the circle in half. A … Read more
Codomain Since you may be dealing with functions and relations unto their depths, knowing better about domain and range sets would be considerable. Codomain of a function The codomain of a function is known to be its set of possible outputs. In other words, codomain is a set of elements that may possibly and logically … Read more