Solve this general form to standard equation of hyperbola;9x²-4y²=36 

To convert the general form of the hyperbola \(9x^2 - 4y^2 = 36\) into its standard form, follow these steps:

1. **Divide the equation by 36** to get it equal to 1:

   \[
   \frac{9x^2}{36} - \frac{4y^2}{36} = 1
   \]
   
   Simplifying, we get:

   \[
   \frac{x^2}{4} - \frac{y^2}{9} = 1
   \]

2. **Identify the standard form** of the hyperbola equation. The standard form of a hyperbola centered at the origin is given as:

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

   Here, we have:

   \[
   \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1
   \]

   So, \(a^2 = 4\) and \(b^2 = 9\), which means:

   \[
   a = 2, \quad b = 3
   \]

3. **Determine the orientation** of the hyperbola. Since the \(x^2\) term has a positive coefficient and comes first, this hyperbola opens left and right.

4. **Summary of the standard form**:
   
   \[
   \frac{x^2}{4} - \frac{y^2}{9} = 1
   \]

   - **Center**: \((0, 0)\)
   - **Vertices**: \((\pm a, 0) = (\pm 2, 0)\)
   - **Co-vertices**: \((0, \pm b) = (0, \pm 3)\)
   - **Foci**: Located at \((\pm c, 0)\), where \(c = \sqrt{a^2 + b^2} = \sqrt{4 + 9} = \sqrt{13}\). So, foci are \((\pm \sqrt{13}, 0)\).

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:

1. How to find the asymptotes of a hyperbola?
2. What are the properties of a hyperbola in standard form?
3. How do you graph a hyperbola given its standard form?
4. How to convert the standard form of a hyperbola to its general form?
5. What is the difference between the equations of a hyperbola and an ellipse?

**Tip:** The value of \( c \) for hyperbolas is calculated using \( c = \sqrt{a^2 + b^2} \), unlike ellipses where it's \( c = \sqrt{a^2 - b^2} \).

Solve this general form to standard equation of hyperbola; 9x² - 4y² = 36

This guide explains how to convert the general form of the hyperbola 9x² - 4y² = 36 to its standard form. Learn about the key algebraic steps, formulas for the vertices, co-vertices, and foci, and the properties of hyperbolas. Suitable for Grades 10-12.

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