Question

For the following exercises, sketch the graph of each conic. $$ 25 x^{2}-4 y^{2}=100 $$

Short answer

The graph is a hyperbola centered at the origin, opening left and right with vertices at \((2,0)\) and \((-2,0)\), and asymptotes with slopes \(\pm \frac{5}{2}\).

Step-by-step solution

Identify the Type of Conic

Given the equation \(25x^2 - 4y^2 = 100\), notice that it can be rewritten in standard form by dividing every term by 100. As a result, the equation becomes \(\frac{x^2}{4} - \frac{y^2}{25} = 1\). Since this equation is of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a^2\) and \(b^2\) are positive, it represents a hyperbola centered at the origin with transverse axis along the x-axis.

Identify the Values of a and b

In the equation \(\frac{x^2}{4} - \frac{y^2}{25} = 1\), we identify \(a^2 = 4\) and \(b^2 = 25\). Solving for \(a\) and \(b\) gives us \(a = 2\) and \(b = 5\). These values determine the distance from the center to the vertices along the x-axis and y-axis, respectively.

Sketch the Axes and Center

Draw the coordinate axes, and mark the center of the hyperbola at \((0,0)\). The center of the hyperbola is the point from which you measure the distances of \(a\) and \(b\).

Determine the Vertices

Since the transverse axis is the x-axis, the vertices of the hyperbola are at \((\pm a, 0)\). Therefore, the vertices are at \((2, 0)\) and \((-2, 0)\). Mark these points on the graph.

Determine the Asymptotes

The asymptotes of the hyperbola can be found using the formula \(y = \pm \frac{b}{a} x\). Given \(a = 2\) and \(b = 5\), the slopes of the asymptotes are \(\pm \frac{5}{2}\). Thus, draw the asymptotes with equations \(y = \frac{5}{2} x\) and \(y = -\frac{5}{2} x\) originating from the center.

Sketch the Hyperbola

The hyperbola will open left and right because the transverse axis is along the x-axis. Use the vertices and asymptotes to guide your sketch. The curves approach but never meet the asymptotes, extending infinitely in both directions.

Key concepts

Hyperbola

A hyperbola is a type of conic section that is formed when a plane intersects both nappes of a double cone. This intersection results in two symmetrical open curves known as branches. Hyperbolas have many unique properties that differentiate them from other conic sections such as ellipses and parabolas.

Key features of a hyperbola include its two branches, each resembling a mirror image of the other, and a center point, which is equidistant from key features like vertices and foci. The branches are the result of the difference of distances from any point on the hyperbola to the two foci being constant. This property aligns with its equation form, where the terms are subtracted.

When visualizing a hyperbola:

• Each branch extends indefinitely outward.
• The vertices are the closest points on each branch to the center of the hyperbola.
• The foci, located further from the center than the vertices, help define the path of the hyperbola.

Hyperbolas have practical applications, including in navigation systems and in the orbits of astronomical objects. Understanding their basics opens a door to diverse mathematical and practical realms.

Standard Form of a Hyperbola

The standard form of a hyperbola is crucial for graphing and understanding its orientation and dimensions. For hyperbolas centered at the origin, this form is expressed as either \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] or \[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \].

The choice between these forms depends on the orientation of the hyperbola's transverse axis. When the transverse axis is horizontal, the first equation is used. This aligns with our current equation after simplification: \[\frac{x^2}{4} - \frac{y^2}{25} = 1 \].

In this equation:

• \(a^2 = 4\) and \(b^2 = 25\)
• The transverse axis runs along the x-axis because the x-term leads the equation.
• "\(a\)" and "\(b\)" represent the distances from the center to the vertices (\(\pm a, 0\)) and the distance from the center to the asymptotes respectively.

This form simplifies understanding the hyperbola's scale and position on a coordinate plane. By altering certain parameters, distinct variations of hyperbolas are explored.

Asymptotes

Asymptotes are imaginary lines that give directionality to the branches of the hyperbola. Though the hyperbola's branches curve towards these lines, they never actually touch them. Asymptotes provide a framework to sketch hyperbolas accurately and predict their behavior at infinity.

The equations of the asymptotes for a hyperbola with its center at the origin, with our given equation \(\frac{x^2}{4} - \frac{y^2}{25} = 1\), can be derived using the formula \(y = \pm \frac{b}{a} x\).

• Here, \(a = 2\) and \(b = 5\), which results in the asymptotes' equations being:
  • \(y = \frac{5}{2} x\)
  • \(y = -\frac{5}{2} x\)

These asymptotes provide guidelines that define the direction in which the hyperbola's branches extend.

To visually capture the relationship of a hyperbola to its asymptotes, graphically show how each branch of the hyperbola remains infinitely close yet parallel to these lines without intersecting them. Understanding asymptotes deepens comprehension of the hyperbola's orientation and how it spans across the coordinate plane.