Problem 68 - Algebra 2

Question

Find the foci of each hyperbola. Draw the graph. \(4 y^{2}-100 x^{2}=400\)

Step-by-Step Solution

Answer

The foci of the hyperbola are at (0, 10.2) and (0, -10.2). By drawing the hyperbola centered at (0,0), with these foci, we get a hyperbola opening upward and downward.

Step 1: Standardise the equation

To bring the equation \(4y^{2} - 100x^{2} = 400\) into the standard form, divide all parts of the equation by 400: \(-\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), we will get \(-\frac{x^{2}}{4} + \frac{y^{2}}{100} = 1\). Here, \(a^{2}=4\) and \(b^{2}=100\). So our 'a' equals to 2, and 'b' equals to 10.

Step 2: Find the foci

Calculate the foci using the relationship \(c = \sqrt{b^{2} + a^{2}}\) where 'c' is the focal distance. Substituting the values in for 'a' and 'b' we get \(c = \sqrt{10^{2} + 2^{2}}\) which simplifies to \(c = \sqrt{104}\) or as a decimal approximately 10.2. The foci of a hyperbola centered at the origin with vertical transverse axis are (0, c) and (0, -c), so our foci are at (0, 10.2) and (0, -10.2).

Step 3: Draw the graph

Draw the hyperbola on the graph using the given equation \(-\frac{x^{2}}{4} + \frac{y^{2}}{100} = 1\). Mark the center at (0,0), then sketch the curve, opening upward and downward. Next, mark points of the foci (0, 10.2) and (0, -10.2) on the y-axis. The foci fall on the vertical axis because of the negative sign associated with the x term.

Key Concepts

Conic Sections

Conic sections are shapes created by the intersection of a plane and a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. The type of conic section formed depends on the angle of the intersecting plane:

• Circle: Formed when the plane cuts parallel to the base of the cone.
• Ellipse: Formed when the plane cuts through at an angle but does not pass through the base.
• Parabola: Occurs when the plane is parallel to a slant edge of the cone.
• Hyperbola: Results when the plane intersects both nappes of the cone.

These shapes have unique properties and equations. Hyperbolas, in particular, are represented by a specific type of quadratic equation, which sets them apart from the other conic sections.

Graphing Hyperbolas

Graphing hyperbolas requires understanding their unique properties. Hyperbolas have two branches that open in opposite directions. Whether these branches open vertically or horizontally depends on the equation's structure.

• In a vertically oriented hyperbola, the equation is of the form \ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \, meaning the hyperbola opens up and down.
• In a horizontally oriented hyperbola, the equation is \ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \, with the branches opening left and right.

To graph a hyperbola:

• Identify the center, typically at \(0,0\) if not shifted.
• Determine the \(a\) and \(b\) values from the equation.
• Plot the vertices and foci based on these values.
• Draw asymptotes, which guide the hyperbola's opening direction.
• Sketch the curve respecting the asymptotes and centered on the vertices.

This step-by-step process helps visualize how the branches of a hyperbola behave.

Finding Foci of Hyperbolas

Finding the foci of hyperbolas is crucial for understanding their shape and properties. The foci are points inside each branch of the hyperbola.

• To find the foci, use the formula \ c = \sqrt{a^2 + b^2} \, where \(a\) and \(b\) come from the equation in standard form.
• The value of \(c\) gives the distance from the center to each focus.

For vertical hyperbolas, the foci are located at \ (0, c) \ and \ (0, -c) \, while for horizontal hyperbolas, they are at \ (c, 0) \ and \ (-c, 0) \. These points are essential in sketching the hyperbola and determining how wide or narrow it is. Foci play a vital role in the geometric definition of hyperbolas as the difference in distances to any point on the hyperbola is constant.

Equation of a Hyperbola

The equation of a hyperbola takes a specific quadratic form. Understanding this form helps identify the hyperbola's orientation and properties.

• The standard form of a hyperbola with a vertical transverse axis is \ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \.
• For a horizontal transverse axis, the standard form becomes \ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \.

In these equations:

• \(a\) represents the distance from the center to each vertex along the transverse axis.
• \(b\) is the distance from the center to each vertex along the conjugate axis.
• The signs in the equations help determine the orientation: negative x-term suggests a vertical orientation, while a negative y-term indicates a horizontal orientation.

To find the foci, use \ c = \sqrt{a^2 + b^2} \. Knowing these equations allows one to properly graph hyperbolas and find important characteristics like the axes, vertices, and foci.