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The semi-major axis of an ellipse is the longest radius from the center to the edge of the ellipse. In simpler terms, it's half the length of the longest line that can be drawn through the center of the ellipse. This concept is crucial in understanding the shape and size of an ellipse. When reading the standard ellipse equation, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the term \(a\) represents the semi-major axis. If \(a > b\), the ellipse stretches more along the x-axis. Conversely, if \(a < b\), it stretches more along the y-axis.
In our given problem, we found \(a^2 = 12\), which means \(a = \sqrt{12} = 2\sqrt{3}\). This tells us how far the ellipse extends horizontally across the x-axis. Understanding the semi-major axis helps in visualizing the overall shape of the ellipse, emphasizing its stretched dimension.

The semi-minor axis is the shortest radius from the center of the ellipse to its edge. Essentially, it's half the length of the shortest line that runs through the center of the ellipse. In the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the term \(b\) stands for the semi-minor axis. It indicates the extent of the ellipse along the y-axis or vertical direction if \(a > b\).
In the exercise, the value of \(b^2\) was determined to be \(9\), giving \(b = \sqrt{9} = 3\). This means that the total vertical height (top to bottom) of the ellipse is twice the semi-minor axis, 6 in this instance. Recognizing \(b\) into the context of the ellipse’s structure allows one to appreciate how the ellipse 'squashes' along the shorter axis, shaping its oval appearance.

The foci (plural for focus) of an ellipse are two special points located along the major axis. The sum of the distances from each focus to any point on the ellipse remains constant. This property is what fundamentally defines an ellipse’s shape.
In our exercise, these foci were at \((\sqrt{3}, 0)\) and \((-\sqrt{3}, 0)\), indicating that the major axis is horizontal. The distance from the center to either focus, denoted as \(c\), is vital. Here, \(c = \sqrt{3}\).
To further grasp this concept, remember that for any ellipse, \(c^2 = a^2 - b^2\). This equation ties the foci with the semi-major and semi-minor axes, offering deeper insight into the ellipse's geometry. In visual terms, if you picture an ellipse as a stretched circle, the foci remain the pivotal points that dictate the extent of that stretch, anchoring the ellipse's structure.

Coordinate geometry, often termed as analytic geometry, involves studying geometric figures using a coordinate system. For ellipses, coordinate geometry provides the tools to represent curves in a plane, with our primary focus on the Cartesian coordinate system.
Through this system, an ellipse can be described via equations that pinpoint its size and shape precisely. The general equation of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) allows us to derive important characteristics like the axes and foci from coordinates accurately.
In solving our ellipse problem, coordinate geometry helps us use set coordinates to define the ellipse's distinctive paths and dimensions. By substituting and solving these coordinates, we navigate smoothly through determining elements like the semi-major and semi-minor axes, reinforcing how algebra and geometry coalesce to unravel such spatial relations.