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The standard form of an ellipse is a fundamental way of expressing the ellipse equation. It helps us describe the shape and properties of an ellipse in a simple form. The standard equation is given by \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where:

• \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively.
• The equation is normalized to equal 1, setting a clear representation of the ellipse based on the axes' lengths.

To rewrite any quadratic ellipse equation into this standard form, you must divide through by the constant term to balance the equation. For instance, in the given problem, we changed \( c x^{2} + 7 y^{2} = 3 \) to a form that resembles the standard formula. We did this by dividing all terms by 3, resulting in \[ \frac{cx^2}{3} + \frac{7y^2}{3} = 1 \] This manipulation simplifies understanding and allows for straightforward computation of axis lengths.

The area of an ellipse is calculated using a simple formula closely related to the lengths of its axes. The formula is \[ \text{Area} = \pi ab \] where:

• \(a\) is the length of the semi-major axis.
• \(b\) is the length of the semi-minor axis.
• \(\pi\) (pi) is a constant roughly equal to 3.14159.

The formula highlights that the area highly depends on the sizes of \(a\) and \(b\). By rearranging the standard form equation, it becomes possible to solve for \(a\) and \(b\) and substitute those values into the area formula. In the exercise, we sought an area of 2. We knew \( \pi ab = 2 \) which enabled us to equate and solve for \(c\) after expressing \(a\) and \(b\) in terms of \(c\) and known values.

Ellipses have two principal axes: the semi-major and semi-minor axes. Understanding their roles is crucial to working with ellipses effectively.

• The **semi-major axis** (\(a\)) is the longest diameter of the ellipse. It stretches from one end of the ellipse to the opposite end, passing through the center.
• The **semi-minor axis** (\(b\)) is perpendicular to the semi-major axis at the center and is the shortest diameter.

From the standard form equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we directly find \(a\) and \(b\) by solving for \(a^2 = \frac{3}{c}\) and \(b^2 = \frac{3}{7}\).These axes are essential in defining not just the shape but also other properties of the ellipse, such as its orientation and eccentricity. In our example, knowing these values allowed us to conclude the equation given results in a certain area, leading to accurate solutions involving the constant \(c\).