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Magazine Chapter 11: Problem 16
Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$\frac{(y-1)^{2}}{25}-(x+3)^{2}=1$$
Short Answer
Expert verified
Center: \((-3,1)\), Vertices: \((-3,6)\), \((-3,-4)\), Foci: \((-3,1\pm\sqrt{26})\), Asymptotes: \(y=5x+16\), \(y=-5x-14\).
Step by step solution
01
Identify the Form of the Hyperbola
The given equation is \(\frac{(y-1)^2}{25} - (x+3)^2 = 1\). This has the form \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\), which represents a vertical hyperbola. Here, \(a^2 = 25\) and \(b^2 = 1\).
02
Determine the Center
For the hyperbola, the center is at the point \((h, k)\). From the equation \(\frac{(y-1)^2}{25} - \frac{(x+3)^2}{1} = 1\), we have \(h = -3\) and \(k = 1\). So, the center is at \((-3, 1)\).
03
Find the Vertices
For a vertical hyperbola, the vertices are \((h, k \pm a)\). Here, \(a = \sqrt{25} = 5\). Thus, the vertices are at \((-3, 1 \pm 5)\) or \((-3, 6)\) and \((-3, -4)\).
04
Find the Foci
The foci of a vertical hyperbola are at \((h, k \pm c)\), where \(c = \sqrt{a^2 + b^2}\). Hence, \(c = \sqrt{25 + 1} = \sqrt{26}\). So, the foci are at \((-3, 1 \pm \sqrt{26})\).
05
Determine the Asymptotes
For a vertical hyperbola, the asymptotes are given by the equations \(y = k \pm \frac{a}{b}(x - h)\). Here, \(\frac{a}{b} = \frac{5}{1} = 5\). So, the equations of the asymptotes are \(y = 1 + 5(x + 3)\) and \(y = 1 - 5(x + 3)\), which simplify to \(y = 5x + 16\) and \(y = -5x - 14\).
06
Sketch the Graph
Plot the center \((-3, 1)\), vertices \((-3, 6)\) and \((-3, -4)\), and draw the asymptotes \(y = 5x + 16\) and \(y = -5x - 14\). Sketch the branches of the hyperbola opening vertically, approaching but never touching the asymptotes.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conic Sections
Conic sections are curves obtained by intersecting a plane with a double napped cone. They are fundamental concepts in mathematics and their study is essential in understanding more advanced geometric properties. The main types of conic sections are:
- Circle - A set of points equidistant from a central point.
- Ellipse - Similar to a circle but stretched along two axes.
- Parabola - A symmetric curve with a single peak or trough.
- Hyperbola - A curve with two separate branches.
Vertices
Vertices are key points on a hyperbola. They represent the points where each branch of the hyperbola is closest to the center. For hyperbolas, there are always two vertices, one on each branch.
- The vertices are located at \((h, k \pm a)\).
Understanding how to find these points is vital as they help in sketching the curve and serve as a guide for locating other key features like the foci and asymptotes.
- In a vertical hyperbola, these vertices lie above and below the center.
- In a horizontal hyperbola, they lie left and right of the center.
- The vertices are located at \((h, k \pm a)\).
Understanding how to find these points is vital as they help in sketching the curve and serve as a guide for locating other key features like the foci and asymptotes.
Foci
The foci (plural of focus) are another set of points that define a hyperbola's properties. These points are located inside each branch of the hyperbola. They play a crucial role in understanding the shape and behavior of the hyperbola.
- For a vertical hyperbola, the foci are positioned vertically relative to the center.
- For a horizontal hyperbola, they are positioned horizontally.
Asymptotes
Asymptotes are lines that a curve approaches as it heads towards infinity.In hyperbolas, these lines provide crucial information about the orientation and the "opening style" of the hyperbola branches.
\(y = k \pm \frac{a}{b}(x-h)\).
This is key for sketching the hyperbola accurately, ensuring you understand how and where the branches approach these lines, especially in a graph.Finding the asymptotes helps create a frame that ensures you draw the hyperbola's shape accurately, reflecting its true geometric properties.
- These lines are straight and do not intercept the hyperbola.
- They usually cross at the hyperbola's center.
\(y = k \pm \frac{a}{b}(x-h)\).
This is key for sketching the hyperbola accurately, ensuring you understand how and where the branches approach these lines, especially in a graph.Finding the asymptotes helps create a frame that ensures you draw the hyperbola's shape accurately, reflecting its true geometric properties.
