Understanding Hyperbola Components

Oct 13, 2025 by leolavanderia.com.br 76 views

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Introduction to Hyperbola

A hyperbola is a type of mathematical curve that consists of two separate branches. It is defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (foci) is constant. In this article, we will focus on finding the equation of a hyperbola with a center at ${$ (0,0)$}$, a vertex at ${$ (0,60)$}$, and a focus at ${$ (0,-65)$}$.

Understanding Hyperbola Components

To represent a hyperbola, we need to understand its key components, which include:

Center: The point at the center of the hyperbola, denoted as ${$ (h,k)$}$.
Vertices: The points on the hyperbola where the curve intersects the transverse axis, denoted as ${$ (h,k \pm a)$}$.
Foci: The points on the hyperbola where the curve intersects the transverse axis, denoted as ${$ (h,k \pm c)$}$.
Transverse Axis: The line segment that passes through the center and the vertices of the hyperbola.
Conjugate Axis: The line segment that is perpendicular to the transverse axis and passes through the center of the hyperbola.

Hyperbola Equation

The standard equation of a hyperbola is given by:

${$ \frac{(x-h)^2}{a2} - \frac{(y-k)^2}{b2} = 1$}$

${$ \frac{(y-k)^2}{a2} - \frac{(x-h)^2}{b2} = 1$}$

where ${$ (h,k)$}$ is the center of the hyperbola, ${$ a$}$ is the distance from the center to the vertices, and ${$ b$}$ is the distance from the center to the co-vertices.

Finding the Equation of the Hyperbola

To find the equation of the hyperbola with a center at ${$ (0,0)$}$, a vertex at ${$ (0,60)$}$, and a focus at ${$ (0,-65)$}$, we need to determine the values of ${$ a$}$ and ${$ c$}$.

From the given information, we can see that the distance from the center to the vertex is ${ $60\$}$ , which is equal to ${$ a$}$. Therefore, ${$ a = 60$}$.

The distance from the center to the focus is ${ $65\$}$ , which is equal to ${$ c$}$. Therefore, ${$ c = 65$}$.

Determining the Value of b

To determine the value of ${$ b$}$, we can use the relationship between ${$ a$}$, ${$ b$}$, and ${$ c$}$:

${$ c^2 = a^2 + b^2$}$

Substituting the values of ${$ a$}$ and ${$ c$}$, we get:

${ $65^2 = 60^2 + b^2\$}$

Simplifying the equation, we get:

${ $4225 = 3600 + b^2\$}$

Subtracting ${ $3600\$}$ from both sides, we get:

${$ b^2 = 625$}$

Taking the square root of both sides, we get:

${$ b = 25$}$

Writing the Equation of the Hyperbola

Now that we have determined the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$, we can write the equation of the hyperbola:

${$ \frac{(x-0)^2}{602} - \frac{(y-0)^2}{252} = 1$}$

Simplifying the equation, we get:

${$ \frac{x^2}{3600} - \frac{y^2}{625} = 1$}$

Conclusion

In this article, we have discussed how to find the equation of a hyperbola with a center at ${$ (0,0)$}$, a vertex at ${$ (0,60)$}$, and a focus at ${$ (0,-65)$}$. We have determined the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ and used them to write the equation of the hyperbola. The equation of the hyperbola is given by:

${$ \frac{x^2}{3600} - \frac{y^2}{625} = 1$}$

This equation represents a hyperbola with a center at ${$ (0,0)$}$, a vertex at ${$ (0,60)$}$, and a focus at ${$ (0,-65)$}$.

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Introduction to Hyperbola Q&A

In our previous article, we discussed how to find the equation of a hyperbola with a center at ${$ (0,0)$}$, a vertex at ${$ (0,60)$}$, and a focus at ${$ (0,-65)$}$. In this article, we will answer some frequently asked questions related to hyperbola equations.

Q: What is the standard equation of a hyperbola?

A: The standard equation of a hyperbola is given by:

${$ \frac{(x-h)^2}{a2} - \frac{(y-k)^2}{b2} = 1$}$

${$ \frac{(y-k)^2}{a2} - \frac{(x-h)^2}{b2} = 1$}$

where ${$ (h,k)$}$ is the center of the hyperbola, ${$ a$}$ is the distance from the center to the vertices, and ${$ b$}$ is the distance from the center to the co-vertices.

Q: How do I determine the values of a, b, and c in a hyperbola equation?

A: To determine the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ in a hyperbola equation, you need to know the coordinates of the center, vertices, and foci of the hyperbola.

The distance from the center to the vertices is equal to ${$ a$}$.
The distance from the center to the foci is equal to ${$ c$}$.
The value of ${$ b$}$ can be determined using the relationship between ${$ a$}$, ${$ b$}$, and ${$ c$}$:

${$ c^2 = a^2 + b^2$}$

Q: What is the relationship between a, b, and c in a hyperbola equation?

A: The relationship between ${$ a$}$, ${$ b$}$, and ${$ c$}$ in a hyperbola equation is given by:

${$ c^2 = a^2 + b^2$}$

This equation shows that the square of the distance from the center to the foci is equal to the sum of the squares of the distances from the center to the vertices and the co-vertices.

Q: How do I write the equation of a hyperbola with a given center, vertex, and focus?

A: To write the equation of a hyperbola with a given center, vertex, and focus, you need to determine the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ using the coordinates of the center, vertices, and foci.

Once you have determined the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$, you can use the standard equation of a hyperbola to write the equation of the hyperbola.

Q: What is the equation of a hyperbola with a center at (0,0), a vertex at (0,60), and a focus at (0,-65)?

A: The equation of a hyperbola with a center at ${$ (0,0)$}$, a vertex at ${$ (0,60)$}$, and a focus at ${$ (0,-65)$}$ is given by:

${$ \frac{x^2}{3600} - \frac{y^2}{625} = 1$}$

This equation represents a hyperbola with a center at ${$ (0,0)$}$, a vertex at ${$ (0,60)$}$, and a focus at ${$ (0,-65)$}$.

Conclusion

In this article, we have answered some frequently asked questions related to hyperbola equations. We have discussed how to determine the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ in a hyperbola equation, the relationship between ${$ a$}$, ${$ b$}$, and ${$ c$}$, and how to write the equation of a hyperbola with a given center, vertex, and focus. We have also provided an example of a hyperbola equation with a center at ${$ (0,0)$}$, a vertex at ${$ (0,60)$}$, and a focus at ${$ (0,-65)$}$.