Determine whether the transverse axis is horizontal or...

Key Concepts

Hyperbola

A hyperbola is a type of conic section defined as the set of all points for which the absolute difference of the distances to two fixed points (the foci) is constant. Its graphical representation consists of two separate curves or branches, and its structure is characterized by its symmetry and orientation determined by its equation.

Standard Forms of Hyperbola

Hyperbolas have standard forms that indicate their orientation. When the equation is written as (x - h)²/a² - (y - k)²/b² = 1, the hyperbola opens horizontally, meaning its branches extend left and right. Conversely, the form (y - k)²/a² - (x - h)²/b² = 1 indicates that the hyperbola opens vertically, with branches extending up and down.

Transverse Axis

The transverse axis is the line segment that connects the vertices of the hyperbola and passes through its center. It determines the direction in which the hyperbola opens. If the equation of the hyperbola places the positive term with the x-variable, the transverse axis is horizontal; if with the y-variable, it is vertical.

Orientation of the Hyperbola

The orientation of the hyperbola is deduced by identifying which variable is associated with the positive term in its standard equation. This identification is key in determining whether the transverse axis is horizontal or vertical, thereby indicating the direction in which the hyperbola's branches open.

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