91Ó°ÊÓ

When studying hyperbolas, one of the tasks is to find the vertices. For a horizontal hyperbola in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the vertices are located at \((\pm a, 0)\).
These points lie along the transverse axis, which is the axis that passes through the center of the hyperbola and is parallel to the x-axis in this case.
Knowing the value of \(a\) is essential, as it helps define how the hyperbola opens, indicating the distances from the center to the vertices along the transverse axis.
In our exercise, after converting the hyperbola’s equation to standard form, we determined \(a^2 = 49\), which gives us \(a = 7\).
So, the vertices are \((7, 0)\) and \((-7, 0)\).
The significance of these points is that they provide a frame for the hyperbola, showing the widths of the 'loops' along the x-axis.

Asymptotes are straight lines that the hyperbola approaches but never meets. They provide a boundary that helps visualize and sketch the hyperbola accurately.
For hyperbolas in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of the asymptotes are given by \(y = \pm \frac{b}{a}x\).
Understanding how asymptotes work allows us to better predict the behavior of the hyperbola as it stretches infinitely far out.
In this exercise, we found \(b^2 = 25\), so \(b = 5\).
The asymptotes of our hyperbola were calculated to be \(y = \pm \frac{5}{7}x\).
This tells us that the hyperbola will closely follow these linear paths without ever crossing or touching them, guiding the hyperbola’s shape and orientation.

The standard form of a hyperbola is crucial for solving many problems related to its graph and properties. For hyperbolas, the standard equation often appears as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
This form tells us whether the hyperbola is oriented horizontally or vertically, based on which term is subtracted.
Transforming any hyperbola equation into this form helps us easily identify key features, such as the vertices and asymptotes.
In the given exercise, we had to divide the entire equation \(25x^2 - 49y^2 = 1225\) by 1225 to achieve this, resulting in \(\frac{x^2}{49} - \frac{y^2}{25} = 1\).
Recognizing this standard form allowed us to conclude that it is a horizontally oriented hyperbola, revealing significant information about its graph.
It streamlines the process of analysis and gives a more straightforward comprehension of the hyperbola's geometrical properties.