91Ó°ÊÓ

Asymptotes are straight lines that a hyperbola approaches but never touches. They give important information about the hyperbola's orientation and shape.
In our example, the asymptotes are given as \( y = \frac{3}{2} x \) and \( y = -\frac{3}{2} x \). These equations indicate that the hyperbola is centered at the origin \( (0,0) \) and the slopes of the asymptotes are \( \pm \frac{3}{2} \). This ratio helps us determine the form of the hyperbola equation.
To find the relationship between the hyperbola's axes, we use the slopes of the asymptotes, which are derived from the ratio \( \frac{b}{a} = \frac{3}{2} \). Solving this equation helps us understand the 'stretch' of the hyperbola in the vertical and horizontal directions.

Vertices are points where the hyperbola intersects its transverse axis. These points provide key information about the size and position of the hyperbola.
For our specific problem, one vertex is given as \( (2,0) \), indicating that \( a = 2 \). The value of \( a \) represents the distance from the center \( (0,0) \) to each vertex along the transverse axis.
Using this vertex information along with the ratio from the asymptotes, we can solve for \( b \) using the relationship \( b = \frac{3}{2} a \). Substituting \( a = 2 \) into the equation gives us \( b = 3 \). This means the distance from the center to where the asymptotes intersect the vertical axis is 3 units.

A hyperbola is a type of conic section formed when a plane intersects both nappes of a double cone. It consists of two disconnected curves called branches that mirror each other.
Hyperbolas have two important axes: the transverse axis and the conjugate axis. The transverse axis lies along the line segment connecting the vertices, while the conjugate axis is perpendicular to the transverse axis at the center point. The point of intersection of these axes is the center of the hyperbola.
The standard form of a hyperbola equation centered at the origin with the transverse axis along the x-axis is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This form tells us about the shape, center, and orientation of the hyperbola.

Hyperbola orientation depends on the position of its transverse axis and conjugate axis.
There are two primary orientations:

• Horizontal: When the transverse axis is horizontal (lies along the x-axis), the equation takes the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
• Vertical: When the transverse axis is vertical (lies along the y-axis), the equation takes the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).

In our example, the asymptotes' slopes \( \pm \frac{3}{2} \) and the vertex \( (2,0) \) indicate a horizontal orientation. This means the equation for our hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Substituting \( a = 2 \) and \( b = 3 \), the final equation is \( \frac{x^2}{4} - \frac{y^2}{9} = 1 \). This form reveals the hyperbola's dimensions and orientation clearly.