Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (0,±2)\(;\) foci: (0,±4)

Step-by-Step Solution

Answer

The standard form of the equation of the hyperbola with the given characteristics is \(\frac{y^2}{4} - \frac{x^2}{12} = 1\).

Step 1: Identify the center, \(a\), and \(c\)

The center of the hyperbola is at the origin (0, 0) because it is the midpoint between the vertices and foci. The distance from the center to the vertices, \(a\), is 2 and the distance from the center to the foci , \(c\), is 4.

Step 2: Calculate \(b^2\)

Use the hyperbola relationship \(c^2 = a^2 + b^2\) to calculate \(b^2\). Substitute \(a = 2\) and \(c = 4\), so it will result in \(b^2 = 4^2 - 2^2 = 12\).

Step 3: Write the equation in standard form

Now substitute \(h = 0\), \(k = 0\), \(a = 2\), and \(b^2 = 12\) into the standard form equation of a hyperbola. The equation will be \(\frac{(y-0)^2}{2^2} - \frac{(x-0)^2}{12} = 1\), which simplifies to \(\frac{y^2}{4} - \frac{x^2}{12} = 1\).

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. When that plane is parallel to the axis of the cone, the result is a parabola. If the plane is perpendicular to the axis, we get a circle. However, if the plane is angled such that it cuts through both nappes of the cone but is not parallel to the axis, the resulting curve is a hyperbola.

A hyperbola can be recognized by its two symmetrical branches, each curving away from the other. They may open upwards and downwards, or left and right, depending on their orientation. Hyperbolas have many fascinating properties and various equations to describe them, one of which is the standard form. Understanding the standard form of a hyperbola is crucial because it reveals important information about the hyperbola's shape and position, such as its vertices, foci, asymptotes, and orientation.

Vertices of a Hyperbola

The vertices of a hyperbola are points where each branch of the hyperbola is closest to the center. These points are crucial as they can help us define the dimensions and orientation of the hyperbola. In the standard form of the equation of a hyperbola, vertices can be found a distance of 'a' units from the center on the transverse axis, which is the axis of symmetry that passes through the vertices. For a vertical hyperbola, the vertices are at \( (h, k \pm a) \) for a hyperbola centered at \( (h, k) \). For a horizontal hyperbola, the vertices would be found at \( (h \pm a, k) \).

In the given exercise, the vertices are at \( (0, \pm 2) \), indicating that the hyperbola is vertical and opens upwards and downwards. The value of 'a' corresponds to the distance from the center to the vertices, which is 2 in this case. This information is instrumental in constructing the hyperbola and distinguishing it from other conic sections.

Foci of a Hyperbola

The foci of a hyperbola are a pair of points located inside each branch, and they are another set of important characteristics defining a hyperbola's shape and equation. The distance from the center to each focus is represented by 'c' in the hyperbola's equations. A unique property of hyperbolas is that for any point on the curve, the difference in distances to the two foci is constant.

The standard form equation of a hyperbola contains the relationship \(c^2 = a^2 + b^2\), where 'b' represents the distance associated with the conjugate axis, perpendicular to the transverse axis. The foci for a vertical hyperbola are located at \( (h, k \pm c) \) and \( (h \pm c, k) \) for a horizontal hyperbola.

In our exercise's context, the foci are at \( (0, \pm 4) \) indicating a larger 'c' value than 'a', which is always the case for hyperbolas. Knowing the foci positions directly contributes to determining the correct form of the hyperbola equation, critical for plotting it accurately and understanding its geometrical properties.