Problem 12 - College Algebra

Question

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

Step-by-Step Solution

Answer

The standard form of hyperbola is \((y−1)^2/9 −(x+2)^2/16 = 1\)

Step 1: Identify the Center, Vertex, and Focus

From the information given, we know that the center of the hyperbola is \((-2,1)\), the focus is \((-2,6)\), and the vertex is \((-2,4)\).

Step 2: Determine Orientation and Calculate 'a'

All points share the same x-coordinate, so the hyperbola opens upwards or downwards. The value of \(a\) is the distance between the center and either vertex, that is \(a = |1-4| =3\).

Step 3: Calculate 'c'

The value of \(c\) is the distance from the center to either focus, hence, \(c = |1-6| =5\).

Step 4: Evaluate 'b' using \(c^2=a^2+b^2\)

Substituting the values of \(a\) and \(c\) into the equation we get, \(b^2 = c^2 - a^2\). Substituting \(a = 3\) and \(c =5\) into the equation, it yields \(b = \sqrt{25 - 9} = 4\).

Step 5: Write the Equation of the Hyperbola

With the center \((-2,1)\), \(a = 3\), \(b = 4\), the standard form of the hyperbola is \((y−1)^2/3^2 −(x+2)^2/4^2 = 1\) which simplifies to \((y−1)^2/9 −(x+2)^2/16 = 1\).

Key Concepts

Conic Sections

The term \textbf{conic sections} refers to the shapes created as a plane slices through a double-napped cone at various angles. These shapes include circles, ellipses, parabolas, and hyperbolas.

Each conic section has its unique properties and equations. For instance, a hyperbola, which is of interest in this exercise, consists of two separate curves that mirror each other. The standard form equation for a hyperbola with vertical transverse axis is \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\] where \(h, k)\) is the center of the hyperbola. The standard form for a hyperbola with a horizontal transverse axis is \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\].

Understanding these forms is crucial for identifying the hyperbola's orientation, which brings us to the main aspects of orientation and its relation to the hyperbola's equation.

Hyperbola Orientation

In analyzing the orientation of a hyperbola, we must look at the structure of its standard form equation. Orientation refers to the direction in which the hyperbola opens. For this particular problem, given that the vertex and focus have the same x-coordinate as the center, we know the hyperbola must open vertically, either upward or downward.

The value of \(a)\) determines the distance between the center and vertices along the orientation axis. Since the given vertex and center share the same x-coordinate, we're dealing with a vertical orientation. This results in the equation's primary fraction involving \(y)\) values, indicating the vertical transverse axis. In simpler terms, if a hyperbola's equation has the \(y)\) term before the \(x)\) term, like in \[\frac{(y−1)^2}{9} − \frac{(x+2)^2}{16} = 1\], it opens up and down.

Distance Formula in Coordinate Geometry

The distance formula in coordinate geometry is pivotal when working with conic sections such as hyperbolas. It allows us to calculate the distance between two points on a coordinate plane. The distance formula is derived from the Pythagorean theorem and is expressed as \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\], where \(x_1, y_1)\) and \(x_2, y_2)\) are coordinates of the two points.

In the context of our hyperbola exercise, we use the distance formula to determine the values of \(a)\) and \(c)\). For example, the distance between the center and the vertex (value \(a)\)) is found by keeping the x-coordinate constant and calculating the difference in y-coordinates. Likewise, the value of \(c)\), the distance between the center and the focus, is calculated similarly since the x-coordinates do not change. These calculations are integral to writing the standard form equation of the hyperbola.