Problem 12 - Algebra and Trigonometry

Question

find the standard form of the equation of each hyperbola satisfying the given conditions. Center: \((-2,1) ;\) Focus: \((-2,6) ;\) vertex: \(\quad(-2,4)\)

Step-by-Step Solution

Answer

The standard form of the equation of the hyperbola is \((x+2)^2/4 - (y-1)^2/9 = 1\)

Step 1: Determine 'a'

First, find the value of 'a', by calculating the distance between the center and the vertex. In this case, the center is (-2,1) and the vertex is (-2,4). The distance between them is 3 units. Therefore, \(a = 3\).

Step 2: Determine 'c'

Next, find the value of 'c', by calculating the distance between the center and the focus. Here, the center is (-2,1) and the focus is (-2,6). The distance between them is 5 units. Therefore, \(c = 5\).

Step 3: Calculate 'b'

The relationship between a, b, and c in a hyperbola is given by \(c^2 = a^2 + b^2\). So to find 'b', substitute 'a' and 'c' into this equation and solve for 'b'. \(b = \sqrt{c^2 - a^2} = \sqrt{5^2 - 3^2} = \sqrt{16}=4\)

Step 4: Formulate the equation

Since the hyperbola is vertical, insert all these values into the standard form equation for a vertical hyperbola, which is \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\). Replace h with -2, k with 1, a with 3 and b with 4 to get the equation \((x+2)^2/4 - (y-1)^2/9 = 1\)

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each type has unique properties and equations that describe its shape and characteristics. Hyperbolas are notably one of these conic sections, characterized by two symmetrical open curves. This results from the plane cutting through both nappes of the double cone.

All conic sections can be represented by a general quadratic equation, but each has its own standard form. Understanding these different equations helps in identifying and graphing the conic sections. Hyperbolas, for instance, are described by equations that differ based on orientation—whether the transverse axis is horizontal or vertical.

Standard Form of Hyperbola

The standard form of a hyperbola's equation depends on the orientation of the transverse axis. Hyperbolas can be either horizontal or vertical. Specifically, the standard form for a vertical hyperbola is

• \[(y-k)^2/a^2 - (x-h)^2/b^2 = 1\]

Here,

• \( (h, k) \) is the center of the hyperbola.
• "a" is the distance from the center to each vertex on the transverse axis.
• "b" is the distance related to the conjugate axis.

For a horizontal orientation, the equation changes to

• \[(x-h)^2/a^2 - (y-k)^2/b^2 = 1\]

These equations reflect the separation into two branches and provide a basis for graphing the hyperbola.

Distance Formula

The distance formula is essential for solving problems involving the geometry of figures, such as finding distances between points related to a hyperbola. It is given by

• \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

This formula calculates the distance between two points in a coordinate plane.

For hyperbolas, it's used to find the distances, \(a\) and \(c\).

• "a" is the distance from the center of the hyperbola to a vertex, and "c" is the distance from the center to a focus.

In the given problem, the formula helps determine these crucial parameters that feed into the hyperbola's equation.

Vertices of Hyperbola

The vertices of a hyperbola are key points that define its shape and orientation. For hyperbolas, the vertices lie on the transverse axis and are located an equal distance from the center of the hyperbola. The distance from the center to a vertex is the value of "a". In the standard equations, this distance is crucial for constructing the hyperbola's equation.

If the equation follows a vertical hyperbola format, the vertices will share the same x-coordinate, while differing by "a" on the y-axis. Conversely, for a horizontal hyperbola, the vertices will share the same y-coordinate and vary on the x-axis.

• For example, in a vertical hyperbola with center \((h, k)\), the vertices are at \((h, k \pm a)\).

Understanding where these vertices are located helps in sketching the correct shape of the hyperbola's graph.