Problem 6 - Algebra and Trigonometry

Question

find the standard form of the equation of each hyperbola satisfying the given conditions. $$ \text { Foci: }(0,-6),(0,6) ; \text { vertices: }(0,-2),(0,2) $$

Step-by-Step Solution

Answer

The standard form of the given hyperbola is \(y^2/4 - x^2/32 = 1\)

Step 1: Identify the Hyperbola Type

Based on the foci and vertices, realize that this is a vertical hyperbola since the points are aligned along the y-axis

Step 2: Calculate Relevant Distances

Determine the distance 'a' from the origin to the vertices which is 2, and the distance 'c' from the origin to the foci which is 6.

Step 3: Calculate 'b' using the relationship between 'a', 'b', and 'c'

In a hyperbola, the relationship between 'a', 'b', and 'c' is given by \(c^2 = a^2 + b^2\). Substitute the values of 'a' and 'c' to solve for 'b'. Square of 'c' is \(6^2=36\) and square of 'a' is \(2^2=4\), therefore, \(b^2 = 36-4 = 32\). So, 'b' equals to \(\sqrt{32}= 4\sqrt{2}\)

Step 4: Write Down the Standard Form of a Hyperbola

The standard forms of a hyperbola are \((x-h)^2/a^2 − (y-k)^2/b^2 = 1\) for a horizontal hyperbola and \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\) for a vertical hyperbola. Here, we are dealing with a vertical hyperbola with center at the origin (0,0). Therefore, the general form becomes \((y-0)^2/2^2 - (x-0)^2/(\sqrt{32})^2 = 1\) which simplifies to \(y^2/4 - x^2/32 = 1\)

Key Concepts

Hyperbola Vertex and Foci

Understanding the relationship between the vertex and foci is fundamental when studying hyperbolas, as these points define the shape and orientation. A hyperbola consists of two vertices, which are the closest points of the hyperbola to its center, and two foci, which are points located inside each branch of the hyperbola at a fixed distance from the center.

In the given exercise, the vertices were identified as (0, -2) and (0, 2), and the foci as (0, -6) and (0, 6). The vertices and foci being aligned along the y-axis indicates that the hyperbola is vertical. The vertex-foci distance for a hyperbola is denoted by 'c', the distance from the center to either vertex is 'a', and the value 'b' relates to 'a' and 'c' through the equation \(c^2 = a^2 + b^2\). When 'a' and 'c' are known, this allows us to solve for 'b', completing the set of parameters necessary to define a hyperbola's standard equation.

Hyperbola Equations

A hyperbola's equation can be identified as belonging to one of two general forms: \(\frac{{(x-h)^2}}{{a^2}} - \frac{{(y-k)^2}}{{b^2}} = 1\) for a horizontal hyperbola, or \(\frac{{(y-k)^2}}{{a^2}} - \frac{{(x-h)^2}}{{b^2}} = 1\) for a vertical hyperbola. 'h' and 'k' denote the coordinates of the hyperbola's center, which is the midpoint between the vertices and the foci.

For our specific problem, since the hyperbola is vertical, the relevant formula is \(\frac{{(y-k)^2}}{{a^2}} - \frac{{(x-h)^2}}{{b^2}} = 1\). After determining that the center is at the origin (0,0), substituting 'h' and 'k' with zeros simplifies our equation to \(\frac{{y^2}}{{4}} - \frac{{x^2}}{{32}} = 1\), which is the standard form of a vertical hyperbola.

Solving for Hyperbola Parameters

Solving for a hyperbola's parameters involves finding the numerical values for 'a', 'b', and 'c' that satisfy the relationship \(c^2 = a^2 + b^2\). From the vertices (0, -2) and (0, 2), the distance 'a' is easily determined as 2, since it represents half the distance between these points. 'c' is identified similarly from the foci (0, -6) and (0, 6), giving us a value of 6.

Using the equation \(c^2 = a^2 + b^2\), we now have enough information to calculate 'b'. In the exercise, by substituting the given values and rearranging, we find that 'b' is the square root of 32, or \(4\sqrt{2}\). Each of these parameters is crucial to outline the hyperbola correctly. With 'a', 'b', and 'c' known, we can articulate the standard form of the hyperbola, sketch its graph, and understand its geometric properties in the context of the coordinate plane.