In Exercises 27–32, find the standard form of the equation of each hyperbola.

Verified step by step guidance

1. Identify the center of the hyperbola, which is at the origin (0,0).
2. Determine the vertices of the hyperbola, which are at (0,7) and (0,-7). This gives us the value of 'a' as 7.
3. Identify the asymptotes of the hyperbola. The slopes of the asymptotes are given by the lines passing through the origin with slopes ±7/5. This gives us the value of 'b' as 5.
4. Write the standard form of the equation of the hyperbola centered at the origin with vertical transverse axis: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
5. Substitute the values of 'a' and 'b' into the equation: \( \frac{y^2}{7^2} - \frac{x^2}{5^2} = 1 \).

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola centered at the origin is given by the equations (x²/a²) - (y²/b²) = 1 or (y²/b²) - (x²/a²) = 1. Here, 'a' represents the distance from the center to the vertices along the x-axis or y-axis, while 'b' represents the distance to the asymptotes. Understanding this form is crucial for identifying the hyperbola's orientation and dimensions.

Asymptotes of a Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola in standard form, the equations of the asymptotes can be derived from the standard form equations. They are given by y = (b/a)x and y = -(b/a)x for hyperbolas centered at the origin. Recognizing the asymptotes helps in sketching the hyperbola accurately and understanding its behavior.

Vertices and Co-vertices

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis, while the co-vertices are associated with the conjugate axis. For a hyperbola in standard form, the vertices are located at (±a, 0) or (0, ±b) depending on the orientation. Identifying these points is essential for graphing the hyperbola and understanding its geometric properties.