Problem 50 - College Algebra

Question

Given information about the graph of the hyperbola, find its equation. Center: \((3,5) ;\) vertex: \((3,11) ;\) one focus: \((3,5+2 \sqrt{10})\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \((y-5)^2/36 - (x-3)^2/4 = 1.\)

Step 1: Determine the orientation of the hyperbola

The center and the vertex have the same x-coordinate, which means the hyperbola is vertical. Therefore, its standard equation is in the form \[(y-k)^2/a^2 - (x-h)^2/b^2 = 1\], where \((h,k)\) is the center.

Step 2: Identify the center of the hyperbola

The center of the hyperbola is given as \((3,5)\). This means \(h = 3\) and \(k = 5\). We will use this information in the standard form of the hyperbola equation.

Step 3: Calculate the distance a (semi-major axis)

The vertex is \((3,11)\), and the center is \((3,5)\), so the distance between the center and the vertex is \(|11 - 5| = 6\). Therefore, \(a = 6\).

Step 4: Calculate the distance c (distance to the focus)

The focus is given as \((3, 5+2\sqrt{10})\). The distance from the center \((3,5)\) to the focus is \(2\sqrt{10}\), hence \(c = 2\sqrt{10}\).

Step 5: Use relationship between a, b, and c

For a hyperbola, the relationship is \(c^2 = a^2 + b^2\). Substitute \(c = 2\sqrt{10}\) and \(a = 6\) to find \(b^2\): \[(2\sqrt{10})^2 = 6^2 + b^2\] \[40 = 36 + b^2\] \[b^2 = 4\].

Step 6: Write the equation of the hyperbola

Substitute \(h = 3\), \(k = 5\), \(a^2 = 36\), and \(b^2 = 4\) into the standard form equation to get: \[(y-5)^2/36 - (x-3)^2/4 = 1\].

Key Concepts

vertical hyperbola

A vertical hyperbola is a type of hyperbola where the primary opening is in the vertical direction. This means that the hyperbola stretches upwards and downward rather than side-to-side.
The distinguishing factor for a vertical hyperbola is that the center and the vertex share the same x-coordinate. In this specific case, both the center and vertex of the hyperbola are aligned along the y-axis at x = 3.
This orientation is critical when determining the equation of the hyperbola, which will follow this pattern:

• The formula for a vertical hyperbola: \[(y-k)^2/a^2 - (x-h)^2/b^2 = 1,\] where \((h, k)\) is the center of the hyperbola.
• Note how the \(a^2\) term is linked to the y-part of the equation, highlighting the vertical nature.

Recognizing these structures early helps in easily writing the appropriate hyperbola equation.

semi-major axis

The semi-major axis in a hyperbola plays a pivotal role in defining its size and shape. For a vertical hyperbola, this axis is aligned vertically, spanning from the center to the vertex.
In simpler terms, if you imagine a line from the center point stretching directly to the top vertex of the hyperbola, that line represents the semi-major axis.
To find the length of the semi-major axis, you measure the distance between the hyperbola's center and its vertex along the line determined by the hyperbola's orientation.

• In this case, the center is at \((3,5)\), and the vertex is located at \((3,11)\).
• Thus, the length is calculated as \(|11 - 5| = 6\), giving us that \(a = 6\).

It's crucial to understand that this distance \(a\) signifies the farthest reach of the hyperbola from its center vertically, impacting the equation's formulation.

distance to the focus

The distance to the focus is another significant measure in understanding a hyperbola's geometric nature. It indicates how far the focal point is from the hyperbola’s center.
This distance is noted by the variable \(c\) in hyperbola equations.
For our current scenario, comprehending how this distance affects the equation is key:

• The focus provided is at the point \((3, 5+2\sqrt{10})\), starting from the center \((3,5)\).
• Thus, the distance calculates as \(2\sqrt{10}\).

Recognizing \(c\) is essential for employing the relationship \(c^2 = a^2 + b^2\), aiding in solving for other unknowns like \(b^2\). As you bridge these variables, this distance helps verify and construct the complete hyperbola equation.

standard form of hyperbola

Writing the equation of a hyperbola involves understanding its 'standard form,' a specific arrangement that clearly outlines the hyperbola's characteristics.
For a vertical hyperbola, the equation typically takes the form: \[(y-k)^2/a^2 - (x-h)^2/b^2 = 1.\]

• The \((h, k)\) values represent the coordinates of the center.
• The \(a^2\) value relates to the semi-major axis, indicating the vertical reach from the center.
• The \(b^2\) value, solved through the relationship \(c^2 = a^2 + b^2\), speaks to the secondary horizontal stretch.

As collected previously, we integrate these values: \(h = 3\), \(k = 5\), \(a^2 = 36\), and \(b^2 = 4\), drafting the hyperbola's equation as \[(y-5)^2/36 - (x-3)^2/4 = 1.\]
Adhering to such a form validates the hyperbola's orientation, center, and dimensions, proving indispensable for graphing and further analyses.