Identifying the Conic Section: The Ellipse 4x^2 - 6y^2 36

The equation 4x^2 - 6y^2 36 represents a conic section, which can be identified by rewriting it in standard form. Let's explore the steps involved in transforming the given equation into its standard form and identifying the type of conic section it represents.

Step 1: Normalize the Equation

The first step in identifying the conic section is to normalize the equation by dividing every term by 36, the common denominator. This step is crucial in revealing the true form of the conic section.

[frac{4x^2}{36} - frac{6y^2}{36} 1]

By simplifying, we obtain:

[frac{x^2}{9} - frac{y^2}{6} 1]

Step 2: Recognize the Standard Form

The equation in this form (frac{x^2}{a^2} - frac{y^2}{b^2} 1) suggests that the given conic section is not an ellipse, but rather a hyperbola. This form indicates a hyperbola that opens along the x-axis.

Step 3: Identify the Values of a and b

The values of (a) and (b) can be derived from the denominators of the terms in the equation.

[a^2 9 quad text{and} quad b^2 6]

Thus:

[a 3 quad text{and} quad b sqrt{6}]

Step 4: Analyze the Hyperbola

Given that the equation is in the form (frac{x^2}{a^2} - frac{y^2}{b^2} 1), it represents a hyperbola centered at the origin (0,0). The foci lie along the x-axis because the coefficient of (x^2) is positive. The distance from the center to each vertex (a) is 3, meaning the vertices are located at ((pm 3, 0)).

The asymptotes of the hyperbola can be derived from the equation and are given by:

[y pm frac{b}{a} x pm frac{sqrt{6}}{3} x]

Conclusion

The conic section represented by the equation 4x^2 - 6y^2 36 is a hyperbola with its center at the origin, vertices at ((pm 3, 0)), and asymptotes given by (y pm frac{sqrt{6}}{3} x).

Additional Insights

For a more detailed analysis, we can further explore the properties of the hyperbola. For example, we can derive the location of the foci, which are positioned at:

((pm c, 0)) where (c sqrt{a^2 b^2} sqrt{9 6} sqrt{15})

The asymptotes provide a graphical representation of the boundaries of the hyperbola as the distance between (x) and (y)-values increases.

Key Takeaways

Understanding the standard form of conic sections is essential for identifying and analyzing the properties of curves. Normalization by dividing the equation by the common denominator helps reveal the exact conic section type. The coefficients of (x^2) and (y^2) terms determine whether a conic section is an ellipse, hyperbola, or parabola.

By following these steps, we can effectively identify and classify conic sections, making them easier to analyze and understand.