Central rectangle

Definition

A central rectangle in the context of a hyperbola is a rectangle aligned with the coordinate axes and centered at the origin, whose sides are parallel to the asymptotes of the hyperbola. It helps visualize the relationship between the hyperbola and its asymptotes.

5 Must Know Facts For Your Next Test

1. The vertices of a central rectangle are located at $(\pm a, \pm b)$, where $a$ and $b$ correspond to the distances from the center to the vertices along each axis.
2. The diagonals of this rectangle coincide with the asymptotes of the hyperbola.
3. For a hyperbola given by $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, $a$ represents half of the length of the horizontal axis, and $b$ represents half of the length of the vertical axis.
4. The area of a central rectangle is $4ab$, representing twice the product of its semi-axes lengths.
5. Understanding central rectangles aids in graphing hyperbolas by providing reference points for sketching their asymptotes.

Review Questions

• What are the coordinates for the vertices of a central rectangle for a hyperbola given by $\frac{x^2}{9} - \frac{y^2}{16} = 1$?
• How do you determine where to draw asymptotes for a given hyperbola using its central rectangle?
• Calculate the area of a central rectangle for a hyperbola described by $\frac{x^2}{25} - \frac{y^2}{9} = 1$.

Related terms

Asymptote: A line that approaches but never touches a curve; in hyperbolas, asymptotes guide their shape and direction.

Conjugate Axis: The line segment perpendicular to the transverse axis, passing through the center with endpoints that touch each co-vertex.

Transverse Axis: The line segment that passes through both foci and vertices in a hyperbola, representing its major axis.