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An ellipse is a curve on a plane that surrounds two focal points, and the sum of the distances to the two focal points from any point on the curve is constant. This feature gives the ellipse its oval shape. The general equation for an ellipse with its center at the origin of the coordinate system is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively.

In our exercise, the given equation is \( 25x^2 + 4y^2 = 100 \) which can be rewritten by dividing both sides by 100 as \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \) to match the general form. Here, it's easy to make mistakes with the denominators; it's vital to remember that they correspond to \( a^2 \) and \( b^2 \) - not \( a \) and \( b \) directly.

The semi-major axis (denoted as \( a \) ) and the semi-minor axis (denoted as \( b \) ) of an ellipse are respectively the longest and shortest radii that span from the center to the perimeter of the ellipse. The length of \( a \) indicates the distance from the center to the farthest point on the curve, while \( b \) measures to the closest point.

Understanding these terms properly is essential for accurately drawing the shape of an ellipse. In the given solution, after simplifying the equation, we found \( a = 10 \) and \( b = 5 \) which indicates a semi-major axis that's twice as long as the semi-minor axis. Remember, the axis corresponding to the larger number in the denominator of the standard ellipse equation is the semi-major axis.

The foci (singular: focus) of an ellipse are two points located along the major axis, equally distant from the center, which are critical to the ellipse's definition and properties. The distance between these foci and the center is marked as \( c \) and can be calculated using the formula \( c = \sqrt{a^2 - b^2} \) where \( a \) is the length of the semi-major axis, and \( b \) is the length of the semi-minor axis.

For our exercise, we compute the foci by substituting \( a = 10 \) and \( b = 5 \) to get \( c = \sqrt{(10)^2 - (5)^2} = \sqrt{75} \) or approximately 8.66. These values are then used to plot the foci on the graph, ensuring they are correctly placed on the major axis. If the semi-major axis is horizontal, the foci are located at \( (\pm c, 0) \) or \( (-8.66, 0) \) and \( (8.66, 0) \) in our case. It's a common error to misplace the foci or misunderstand their relation to the ellipse's axes, so double-check your calculations and the orientation of the axes.