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Intercepts are the points where a graph crosses the x-axis and y-axis. To find these points, we set either the x or y variable to zero in the given equation and solve for the other variable. In the equation \(25x^{2} + 4y^{2} = 100\), setting \(y = 0\) gives us the x-intercepts. By doing this, we get \(25x^{2} = 100\), which simplifies to \(x = \pm 2\). Therefore, the x-intercepts are \((2, 0)\) and \((-2, 0)\).

Similarly, setting \(x = 0\) gives us the y-intercepts. By doing this, we get \(4y^{2} = 100\), which simplifies to \(y = \pm 5\). Therefore, the y-intercepts are \((0, 5)\) and \((0, -5)\).

This method works for any conic section, making it a powerful tool in coordinate geometry.

Symmetry in mathematics helps us understand the balance and structure of geometric figures. There are different tests for symmetry about various axes and the origin.

For symmetry about the y-axis, we replace \(x\) with \(-x\) in the equation and check if it remains unchanged. In our equation \(25x^{2} + 4y^{2} = 100\), replacing \(x\) with \(-x\) keeps the equation the same.

Similarly, for symmetry about the x-axis, we replace \(y\) with \(-y\), and our equation remains unchanged. Lastly, for symmetry about the origin, both \(x\) and \(y\) are replaced with \(-x\) and \(-y\) respectively, and if the equation remains unchanged, the graph is symmetric about the origin.

Finding such symmetries can greatly simplify the graphing process and help understand the geometric properties of the equation.

Coordinate geometry, also known as analytic geometry, uses coordinates to represent geometric figures. This allows us to apply algebra to solve geometric problems.

In coordinate geometry, the position of any point can be defined using an ordered pair of numbers, typically \((x, y)\). The given equation \(25x^{2} + 4y^{2} = 100\) is an example of an ellipse in coordinate geometry. We analyze the properties of this ellipse by solving algebraic equations involving these coordinates.
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By understanding the intercepts and the symmetry of the ellipse, we can draw the graph accurately and comprehend its structure better. This method combines algebraic techniques with geometric understanding, making it easier to handle complex shapes and figures.

An ellipse is a type of conic section, formed by the intersection of a plane with a cone. It has two main axes: the major axis and the minor axis. The equation \(25x^{2} + 4y^{2} = 100\) represents an ellipse.

To understand its shape, let's rewrite the equation in the standard form of an ellipse, \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). By dividing the given equation by 100, we get \(\frac{x^{2}}{4} + \frac{y^{2}}{25} = 1\). Here, \(a^{2}\) is 4 and \(b^{2}\) is 25, thus \(a = 2\) and \(b = 5\).

This tells us that the ellipse is stretched more along the y-axis (vertical direction) than along the x-axis (horizontal direction). Knowing the values of 'a' and 'b' helps in accurately sketching the ellipse and understanding its geometric properties.