How to Find the Coordinates of Vertices, Foci, and Equations of Asymptotes for a Hyperbola

In this article, we will guide you through the process of determining the coordinates of the vertices and foci, as well as the equations of the asymptotes for the given hyperbola equation. This involves identifying the standard form of the equation and applying specific formulas.

Step 1: Identify the Standard Form

The standard form of a hyperbola centered at ( (h, k) ) is given by:

[ frac{(x - h)^2}{a^2} - frac{(y - k)^2}{b^2} 1 ]

For the given equation:

[ x - 3^2/4 - y - 1^2/9 1 ]

We can rewrite it as:

[ frac{(x - 3)^2}{4} - frac{(y - 1)^2}{9} 1 ]

From this, we can identify the following:

h 3 k 1 a^2 4 ( Rightarrow ) a 2 b^2 9 ( Rightarrow ) b 3

Step 2: Find the Vertices

The vertices of a hyperbola are located at ( (h pm a, k) ).

Substituting the values, we get:

Vertex 1: ( (3 - 2, 1) (1, 1) ) Vertex 2: ( (3 2, 1) (5, 1) )

Therefore, the vertices are:

( (1, 1) ) and ( (5, 1) )

Step 3: Find the Foci

The foci of a hyperbola are located at ( (h pm c, k) ), where ( c ) is calculated using the formula:

[ c sqrt{a^2 b^2} ]

Substituting the values:

[ c sqrt{4 9} sqrt{13} ]

Thus, the foci are:

F1: ( (3 - sqrt{13}, 1) ) F2: ( (3 sqrt{13}, 1) )

Step 4: Find the Equations of the Asymptotes

The equations of the asymptotes for a hyperbola in this form are given by:

[ y - k pm frac{b}{a} (x - h) ]

Substituting the values, we get:

[ y - 1 pm frac{3}{2} (x - 3) ]

This leads to two equations:

( y - 1 frac{3}{2} (x - 3) ) ( Rightarrow ) ( y - 1 frac{3}{2} x - frac{9}{2} ) ( Rightarrow ) ( y frac{3}{2} x - frac{7}{2} ) ( y - 1 -frac{3}{2} (x - 3) ) ( Rightarrow ) ( y - 1 -frac{3}{2} x frac{9}{2} ) ( Rightarrow ) ( y -frac{3}{2} x frac{11}{2} )

Summary of Results

Vertices: ( (1, 1) ) and ( (5, 1) ) Foci: ( (3 - sqrt{13}, 1) ) and ( (3 sqrt{13}, 1) ) Asymptotes: ( y frac{3}{2} x - frac{7}{2} ) and ( y -frac{3}{2} x frac{11}{2} )

Conclusion

By following these steps, you can easily determine the key characteristics of any hyperbola given its equation. Understanding how to find the vertices, foci, and asymptotes is crucial for analyzing the behavior and properties of hyperbolas.

Remember, practice is the key to mastering these concepts. Keep solving similar problems to reinforce your understanding and improve your skills.