Problem 13 - Algebra and Trigonometry

Question

use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$ \frac{x^{2}}{9}-\frac{y^{2}}{25}=1 $$

Step-by-Step Solution

Answer

The hyperbola \( \frac{x^{2}}{9}-\frac{y^{2}}{25}=1 \) has its vertices at (±3,0), foci at (±√34,0), and asymptotes given by the equations y=± \(\frac{5x}{3}\).

Step 1: Identify the Vertices

For a hyperbola, the vertices are located at (±a, 0) when center is at the origin. Here, the given equation is \( \frac{x^{2}}{9}-\frac{y^{2}}{25}=1 \), so a^2=9. Taking square root on both sides, we get a=±3. So, the vertices are located at (±3, 0).

Step 2: Calculate the Foci

The foci of a hyperbola is given by the equation \(c=\sqrt{a^{2}+b^{2}}\). Here, a^2=9 and b^2=25. Hence, c= ± √(9+25) = ± √34. Therefore, the coordinates of the foci are (±√34, 0).

Step 3: Determine the Equations of Asymptotes

The equations for asymptotes of a hyperbola in this form are given by \(y=±\frac{b}{a}x\). Here, a=3 and b=5. Therefore, the equations of the asymptotes are y=± \(\frac{5x}{3}\)

Step 4: Plot the Graph

Start by drawing the center at the origin. Plot the vertices and foci along the x-axis. Draw the two branches of the hyperbola, opening left and right. The foci should be interior points of each branch. Sketch the asymptotes as diagonals bisecting the rectangle formed by vertices and coordinate axes. They should pass through the center, creating a guide within which the branches of the hyperbola will approach but never reach or cross.

Key Concepts

Vertices of Hyperbola

The vertices of a hyperbola are crucial in determining its shape and orientation. For the given hyperbola equation \( \frac{x^{2}}{9}-\frac{y^{2}}{25}=1 \), the vertices of the hyperbola can be found using the formula for a hyperbola centered at the origin: the vertices are situated at \((\pm a, 0)\) when the hyperbola opens horizontally. Here, \( a^{2} = 9 \), so \( a = \pm 3 \). Thus, the vertices are located at \((3, 0)\) and \((-3, 0)\). These points are at the same distance on either side of the center along the x-axis.

Vertices act as points where each branch of the hyperbola intersects its transverse axis. The transverse axis is the line segment connecting the two vertices. By marking these points on the graph, you begin to create the form and direction of the hyperbola's branches.

Asymptotes of Hyperbola

Asymptotes are lines that the hyperbola approaches but never actually touches or crosses. They provide a framework for sketching the hyperbola because its branches will get infinitely close to the asymptotes as they extend outward.

For a hyperbola in the form of \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), the asymptotes' equations are given by \( y = \pm \frac{b}{a}x \). In our equation \( \frac{x^{2}}{9}-\frac{y^{2}}{25}=1 \), \( a = 3 \) and \( b = 5 \). Therefore, the equations for the asymptotes are \( y = \pm \frac{5}{3}x \).

• Asymptotes form the diagonals of a rectangle centered at the hyperbola's center.
• The branches of the hyperbola will hover around these asymptotes asymptotically, never intersecting them.

These lines are essential for guiding how the hyperbola is drawn, helping ensure its branches curve correctly within the specified framework.

Foci of Hyperbola

The foci of a hyperbola are points located along its transverse axis, beyond the vertices. They are pivotal in defining the hyperbola's shape through the difference in distances from any point on the hyperbola to each focus, which remains constant. In our equation \( \frac{x^{2}}{9} - \frac{y^{2}}{25} = 1 \), foci calculations are rooted in the relationship \( c = \sqrt{a^{2} + b^{2}} \). Given \( a^2 = 9 \) and \( b^2 = 25 \), this leads to \( c = \sqrt{34} \). The foci are therefore located at \((\pm \sqrt{34}, 0)\).

These points are integral to the properties and graphing of the hyperbola, providing interior coordinates that the branches of the hyperbola stretch towards but never reach.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone, and they include circles, ellipses, parabolas, and hyperbolas. Hyperbolas are characterized by their two distinct, opposite-opening branches and arise when the cutting plane intersects both napps of the cone at an angle. This results in the unique properties and structure of a hyperbola.

• Each of these conic sections has a distinct standard form equation.
• Hyperbolas, such as the one given by \( \frac{x^{2}}{9} - \frac{y^{2}}{25} = 1 \), consist of two separate curves.
• They have vertices, foci, and asymptotes, distinguishing them from other conics through these properties.

Understanding conic sections as a whole aids in grasping the mathematical and geometrical relationships that define hyperbolas, offering insight into their unique behavior among conics.