Math Problem Statement

,b, and c. ((x^2)/4)-(y^2)=1

Solution

The equation you've provided resembles a hyperbola. Let's rewrite and analyze the equation:

x24y2=1\frac{x^2}{4} - y^2 = 1

Analysis

  1. Standard Form: The equation of a hyperbola in standard form is given by:

    x2a2y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

    In this form, a2a^2 is associated with the xx-term, and b2b^2 with the yy-term if the hyperbola opens left-right, or vice versa if it opens up-down.

  2. Identification of a2a^2 and b2b^2: Comparing your equation with the standard form:

    • a2=4a^2 = 4 so a=2a = 2
    • b2=1b^2 = 1 so b=1b = 1
  3. Characteristics:

    • The hyperbola opens left-right because the xx-term is positive.
    • The transverse axis length is 2a=42a = 4.
    • The conjugate axis length is 2b=22b = 2.
    • The center of the hyperbola is at the origin (0,0)(0,0).

Related Questions

  1. What is the length of the transverse axis?
  2. How would the equation change if the hyperbola opened up-down?
  3. Can you find the equations of the asymptotes for this hyperbola?
  4. What are the coordinates of the vertices of the hyperbola?
  5. How would the equation change if the center of the hyperbola were at (h,k)(h, k)?
  6. How do you find the foci of this hyperbola?
  7. Can you sketch the graph of this hyperbola?
  8. How does the equation change if the hyperbola is rotated?

Tip

To determine the orientation of a hyperbola, check the signs in the standard form equation. If the xx-term is positive, the hyperbola opens horizontally; if the yy-term is positive, it opens vertically.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Hyperbola
Conic Sections

Formulas

Standard form of a hyperbola

Theorems

-

Suitable Grade Level

Advanced High School