Understanding the Conic Section of 25x^2-9y^2225: Foci and Properties

Dealing with conic sections can sometimes be quite complex, but the process often simplifies when we follow the right steps. One of the fascinating conic sections is the hyperbola, and this article will guide you through the process of determining the foci of the given equation, 25x^2-9y^2225.

Standard Form Transformation

Let's start by transforming the given equation into its standard form. The initial equation is:

25x^2 - 9y^2 225

To achieve the standard form, we need to divide every term by 225, the constant term on the right side of the equation. This will give us:

(frac{25x^2}{225} - frac{9y^2}{225} frac{225}{225})

Simplifying further, we get:

(frac{x^2}{9} - frac{y^2}{25} 1)

Identifying the Conic Section and Key Parameters

Now, let's analyze this equation in the context of conic sections. The standard form of a hyperbola is:

(frac{x^2}{a^2} - frac{y^2}{b^2} 1)

Comparing the transformed equation with the standard form, we can identify that:

(a^2 9) and (b^2 25)

This tells us that the conic section is indeed a hyperbola.

Determining the Foci

Next, we need to find the foci of this hyperbola. The formula to find the distance of the foci from the center (0, 0) is given by:

c^2 a^2 b^2

Substituting the values of a^2 and b^2, we get:

c^2 9 25

Therefore:

c^2 34

And:

c (sqrt{34})

The foci are located at ((pm c), 0). Since (a^2 9) (which is positive), the major axis is along the x-axis. Therefore, the coordinates of the foci are:

(-(sqrt{34}), 0) and ((sqrt{34}), 0)

Additional Insights

The hyperbola equation (frac{x^2}{9} - frac{y^2}{25} 1) also provides us with the following properties:

Center of the Hyperbola: The center is located at (0, 0). Vertices: The vertices are located at ((pm a), 0), which are ((pm 3), 0). Asymptotes: The equations of the asymptotes for this hyperbola are given by (y pm frac{b}{a}x), which simplifies to (y pm frac{5}{3}x).

Conclusion

While the process of identifying the foci of a conic section can involve several steps, understanding the standard form and the key parameters significantly simplifies the task. In the case of the hyperbola 25x^2-9y^2225, we found that the foci are at ((-sqrt{34}, 0)) and ((sqrt{34}, 0)), and we identified several important properties of the hyperbola.

By mastering these concepts, you can confidently handle more complex problems in conic sections and expand your knowledge in this area.