Determine the equations in standard form of two different hyperbolas that satisfy the given cond

Determine the equations in standard form of two different...

Key Concepts

Asymptotes

Asymptotes are lines that a hyperbola approaches but never touches. For hyperbolas with a horizontal transverse axis, the equations of the asymptotes are given by y = k ± (b/a)(x - h). They provide a framework that helps in graphing the hyperbola and understanding the relationship between the parameters a and b, especially through the slope (b/a) which is critical when certain asymptotic conditions are provided.

Hyperbola

A hyperbola is a type of conic section that is defined as the set of all points where the absolute difference of the distances from two fixed points (the foci) is constant. It consists of two disconnected curves called branches, which open in opposite directions. Understanding hyperbolas involves recognizing their geometric properties, such as vertices, foci, axes, and asymptotes, which are all interrelated.

Standard Form Equation

Hyperbolas can be expressed in a standard form equation. For a hyperbola that opens left and right (horizontal transverse axis), the equation is written as (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1, where (h, k) is the center. This form highlights the distances from the center to the vertices (a) and the relationship with the distance parameter (b) that determines the steepness of the asymptotes.

Transverse Axis

The transverse axis of a hyperbola is the line segment that connects the two vertices. Its length is 2a, and its orientation (horizontal or vertical) determines the structure of the hyperbola’s standard form equation. The transverse axis is crucial because it sets the direction in which the branches open and determines the positions of the vertices relative to the center.

Vertices

Vertices are the points where the hyperbola intersects its transverse axis. They are positioned at a distance of a units from the center along the transverse axis. Knowing one vertex allows you to relate it to the center by considering the distance a, which is half the length of the transverse axis, thereby guiding the construction of the hyperbola’s equation.

Center

The center of a hyperbola is the midpoint of the line segment joining the vertices, and it is denoted as (h, k) in the standard equation. It serves as a key reference point for the hyperbola’s geometry, influencing the positions of the vertices, asymptotes, and the overall symmetry of the graph. Determining the center is fundamental when constructing the standard form equation from given geometric details.

Transcript

00:01 Okay, so given this information, we're going to compose two different ways we can write the standard form of a hyperbola.

00:07 So first, i know that my hyperbola is going to have a horizontal transverse axis, so i know it's going to be written as x minus h the quantity squared over a squared minus y minus k.

00:30 That whole quantity squared over b squared, and i know it's going to be set equal to one.

00:38 Okay.

00:40 So i know it's going to be written in this standard form.

00:44 From the information given, though, we can write two possible scenarios.

00:48 So first, the transverse length is 12.

00:51 So the distance between the two vertices is going to be 12.

00:56 And i know that in between those vertices in the middle is the center.

01:00 So, in order to find the distance from one vertex to the center, i could just cut 12 and half, and that'll be 6.

01:09 And i know that the distance from the center to the vertex is measured by a.

01:13 So i know that my a value is going to be 6.

01:19 From there, it's a horizontal transverse axis.

01:25 So i know that it's going to look like, it's going to look like so with the center.

01:36 So i know that one of my vertices is going to be at 6 .5, and the other is going to be at another point.

01:47 But i don't know, this is only one of my vertices, and i don't know if this is the one where i subtract 6 from the x value or add 6.

01:54 So i have two possible scenarios here.

01:58 My first scenario is that i'm adding 6 to the vertex...

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