Problem 10 - Algebra and Trigonometry

Question

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ \frac{y^{2}}{9}-\frac{x^{2}}{16}=1 $$

Step-by-Step Solution

Answer

Vertices: (0, ±3), Foci: (0, ±5), Asymptotes: y = ±(3/4)x.

Step 1: Identify the Hyperbola Form

The given equation \( \frac{y^2}{9} - \frac{x^2}{16} = 1 \) is in the standard form of a hyperbola centered at the origin \( (0, 0) \) with the equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). From this, we can see that \( a^2 = 9 \) and \( b^2 = 16 \). Thus, \( a = 3 \) and \( b = 4 \).

Step 2: Find the Vertices

Since the hyperbola is vertical, the vertices are at \( (0, \pm a) \). Substituting the value of \( a \), we get the vertices as \( (0, 3) \) and \( (0, -3) \).

Step 3: Determine the Foci

For a hyperbola of the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the distance to the foci from the center is given by \( c = \sqrt{a^2 + b^2} \). Substituting \( a^2 = 9 \) and \( b^2 = 16 \), we find \( c = \sqrt{9 + 16} = \sqrt{25} = 5 \). Thus, the foci are at \( (0, 5) \) and \( (0, -5) \).

Step 4: Find the Asymptotes

The asymptotes for a vertical hyperbola \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) are given by the equations \( y = \pm \frac{a}{b}x \). With \( a = 3 \) and \( b = 4 \), the equations of the asymptotes are \( y = \pm \frac{3}{4}x \).

Step 5: Sketch the Hyperbola

In the sketch of the hyperbola, mark the center at the origin, the vertices at \( (0, 3) \) and \( (0, -3) \), and the foci at \( (0, 5) \) and \( (0, -5) \). Draw the asymptotes as lines passing through the origin with slopes \( \frac{3}{4} \) and \( -\frac{3}{4} \), and sketch the branches of the hyperbola opening upwards and downwards between these asymptotes.

Key Concepts

Vertices of Hyperbola

In the context of a hyperbola, the vertices are very special points that are located on the principal axis, at the extremities of the hyperbola's 'opening'. To find these, you need to understand the orientation of the hyperbola. For vertical hyperbolas, like the one given by the equation \( \frac{y^2}{9} - \frac{x^2}{16} = 1 \), the vertices are placed along the y-axis.
Since the equation is in the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the value of \(a\) represents the distance from the center to each vertex along the y-axis:

• This distance is given by \(a = 3\), where \(a\) is derived from \(a^2 = 9\).
• Thus, the vertices are at \((0, \pm 3)\), which are \((0, 3)\) and \((0, -3)\).

These points are crucial because they define how wide the hyperbola opens. Always remember, the vertex points are directly linked to the parameter \(a\) found in your equation.

Foci of Hyperbola

The foci (singular: focus) of a hyperbola are a pair of points located along the principal axis, further out than the vertices. They help in defining the shape and size of the hyperbola. For the vertical hyperbola \( \frac{y^2}{9} - \frac{x^2}{16} = 1 \), finding the foci means using the formula related to \(a\), \(b\), and a new parameter \(c\):

• First, calculate \(c\) using \( c = \sqrt{a^2 + b^2} \).
• Substitute \(a^2 = 9\) and \(b^2 = 16\) into the equation to get \(c = \sqrt{25} = 5\).
• Therefore, the foci are at \((0, \pm 5)\), which are \((0, 5)\) and \((0, -5)\).

Foci stretch the hyperbola, ensuring its two parts are appropriately shaped. They are central to understanding the hyperbola's geometry.

Asymptotes of Hyperbola

Asymptotes of a hyperbola are straight lines that the curve approaches but never actually reaches. These lines act as guidelines, showing the direction in which the hyperbola opens. Understanding them provides a foundation for sketching the curve. For a vertical hyperbola like \( \frac{y^2}{9} - \frac{x^2}{16} = 1 \), the equations of the asymptotes use the values of \(a\) and \(b\):

• They are given by the formula \( y = \pm \frac{a}{b}x \).
• In this case, \(a = 3\) and \(b = 4\), so the asymptotes become \(y = \pm \frac{3}{4}x\).

When sketching, draw these lines as dashed lines through the origin and extending infinitely. They help frame the hyperbola's shape, with the curve "hovering" near them in the positive and negative directions along the x-axis. By understanding asymptotes, you gain insight into the hyperbola's infinite extension.