Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis o

Find the center, the vertices, the foci, the asymptotes, and...

Key Concepts

Rotation of Axes

Rotation of axes is a coordinate transformation technique used to eliminate the cross-product term in conic sections’ equations that contain an xy term. By rotating the coordinate system by an appropriate angle, the hyperbola’s equation can be rewritten in a standard form, making its geometric properties easier to identify and analyze.

Asymptotes

Asymptotes are straight lines that the hyperbola approaches but never intersects, serving as a guide for the behavior of the curve as the absolute values of the coordinates increase. They provide a framework that helps in sketching and understanding the overall orientation and spread of the hyperbola.

Foci

Foci (plural of focus) are fixed points located along the transverse axis of a hyperbola, outside the vertices, that determine the shape of the curve. The defining geometric property of a hyperbola is that the absolute difference of the distances from any point on the hyperbola to these two foci is constant.

Transverse Axis

The transverse axis is the line segment that passes through the center and connects the vertices of the hyperbola. This axis determines the main direction of the hyperbola’s opening and plays a critical role in defining its standard equation.

Center

The center of a hyperbola is the point that serves as the midpoint between its vertices and is the intersection point of its asymptotes. It represents the point of symmetry for the hyperbola, and many of its features are described relative to this point in the standard form equation.

Vertices

Vertices are the points on a hyperbola that are closest to the center along the transverse axis. They define the ‘turning points’ of the branches and are used to help determine the hyperbola’s orientation and scale, with the distance from the center to a vertex being a key parameter in the equation.

Hyperbola

A hyperbola is a type of conic section formed by the intersection of a plane and a double-napped cone, characterized by two disconnected branches that are mirror images of each other. Its standard form, once properly oriented, clearly exhibits properties such as a center, vertices, foci, and asymptotes. The definition and properties of a hyperbola provide insight into its geometric and algebraic structure.

Transcript

00:02 In this question, an equation of a hyperbola is given as x times y equals to 1.

00:06 We need to find out the coordinates of center, vertices and foci of this hyperbola, and the equations of asymptotes and the length of transverse axis of this hyperbola.

00:21 So, first of all, we will convert this equation into another cartesian system having as the capital x, y, system such as the small x y system will have the relation with that as small x x equals to capital x plus capital y and small y will be equal to capital x minus capital y it is also given as hint in this question so now converting this system to another coordinate system that is x y system xy system xy coordinate system so we will get this equation of this hyperbola as that is x equal to x plus y capital x plus y so this will become capital x plus y times capital x minus y that is small y is equal to capital x minus y will be equal to one and we will have it as x square minus y square equal to one we can write it as x square over one minus y square over one equal to one now comparing it to the that is general equation of hyperbola which is x minus h whole square over a square minus y minus k whole square over b square equal to one so comparing this equation the given equation to this equation we will get the values of h k a and b so the value of h will be 0 the value of k will be again zero the value of a is 1 the value of b is also 1 we can write now we can write the coordinates of the center vertices focii use in the system that is capital x capital y.

02:21 So in capital x capital y system the coordinates of center, coordinate of center will be as h comma k.

02:33 Here h is 0 and k is also 0 so therefore the coordinates of center will be 0 .0...

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