Math Problem Statement

What is the standard form equation of the hyperbola that has vertices (0,±2) and foci (0,±27–√)

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Enter the exact equation.

Include a multiplication sign between symbols. For example, x*5–√ .

Solution

To determine the equation of the hyperbola, let's analyze the given information step by step.

Step 1: Identify the type of hyperbola

The hyperbola has vertices at (0,±2)(0, \pm 2), meaning it is centered at (0,0)(0, 0) and opens along the yy-axis. This makes it a vertical hyperbola.

The standard form of a vertical hyperbola is: y2a2x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Step 2: Find a2a^2

The distance from the center to the vertices is aa. Here, a=2a = 2, so: a2=22=4a^2 = 2^2 = 4

Step 3: Use the distance between the foci to find c2c^2

The distance from the center to the foci is cc. Here, c=27c = \sqrt{27}, so: c2=(27)2=27c^2 = (\sqrt{27})^2 = 27

Step 4: Find b2b^2 using the relationship c2=a2+b2c^2 = a^2 + b^2

The relationship between aa, bb, and cc in a hyperbola is: c2=a2+b2c^2 = a^2 + b^2 Substitute the known values: 27=4+b227 = 4 + b^2 Solve for b2b^2: b2=274=23b^2 = 27 - 4 = 23

Step 5: Write the equation

Substitute a2=4a^2 = 4 and b2=23b^2 = 23 into the standard form equation: y24x223=1\frac{y^2}{4} - \frac{x^2}{23} = 1

Final Answer:

The standard form of the hyperbola is: y24x223=1\frac{y^2}{4} - \frac{x^2}{23} = 1

Would you like further details or clarifications?


Related Questions:

  1. How do you derive the standard form of a hyperbola?
  2. What are the equations of asymptotes for this hyperbola?
  3. How do you graph a hyperbola given its equation?
  4. Can you find the eccentricity of this hyperbola?
  5. What is the significance of aa, bb, and cc in the hyperbola's equation?

Tip:

Always verify whether the hyperbola opens horizontally or vertically before using its standard

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Conic Sections
Hyperbolas

Formulas

Standard form of a vertical hyperbola: (y^2 / a^2) - (x^2 / b^2) = 1
Relationship: c^2 = a^2 + b^2

Theorems

Definition of Hyperbolas
Distance Formula for Conic Sections

Suitable Grade Level

Grades 10-12