Question

Find the area of the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\).

Short answer

The area of the ellipse is \(\pi ab\).

Step-by-step solution

Recognize the Standard Form of an Ellipse

The given equation of the ellipse is \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\). Rearrange this equation to match the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Divide everything by \(a^2 b^2\) to get this form.

Divide by Constant to Simplify Equation

Divide each term in the equation \(b^2 x^2 + a^2 y^2 = a^2 b^2\) by \(a^2 b^2\). The equation becomes \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Identify Semi-Axes Lengths

In the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the denominator of the \(x^2\) term, \(a^2\), indicates that \(a\) is the semi-major axis length if \(a > b\), and the denominator of the \(y^2\) term, \(b^2\), shows that \(b\) is the semi-minor axis length. Here, \(a\) and \(b\) are interchangeable as the naming depends on their relative lengths.

Use the Ellipse Area Formula

The area \(A\) of an ellipse is given by the formula \(A = \pi ab\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively.

Calculate the Area

Since \(a\) and \(b\) are already given in the problem without specific values or ordering constraints, calculate the area using \(A = \pi ab\). The area of the ellipse is \(\pi \times a \times b = \pi ab\).

Key concepts

Ellipse Standard Form

An ellipse is a symmetric shape similar to a flattened circle. Its equation can be expressed in a standard form, which makes it easier to identify and work with. The standard form of an ellipse's equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This format helps in pinpointing the lengths of the semi-major and semi-minor axes. By rearranging the given ellipse equation \(b^2 x^2 + a^2 y^2 = a^2 b^2\) into this standard form, you can better understand and calculate its properties. To convert an equation to this form, divide each term by \(a^2 b^2\), allowing us to clearly distinguish the roles of \(a\) and \(b\). This conversion forms the foundation for solving and analyzing ellipses.

Semi-Major Axis

The semi-major axis of an ellipse is the longest radius that extends from the center to the edge of the ellipse. In the standard form equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the larger of the two denominators corresponds to the square of the semi-major axis.

### Determining the Semi-Major Axis- If \(a > b\), then \(a\) is the length of the semi-major axis.- Conversely, if \(b > a\), the semi-major axis will be \(b\).
This axis is crucial in calculating properties of the ellipse, including its area, and affects how stretched out the ellipse is along one axis. Whether it is aligned with the x-axis or y-axis depends on which variable's associated denominator is larger.

Semi-Minor Axis

Conversely, the semi-minor axis is the shorter radius extending from the center to the boundary of the ellipse. In the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the smaller denominator indicates the square of the semi-minor axis.

### Determining the Semi-Minor Axis- If \(a < b\), then \(a\) is the length of the semi-minor axis.- Similarly, if \(b < a\), the semi-minor axis will be \(b\).
This axis affects the compactness of the ellipse along the shorter dimension and is also essential in calculating the ellipse's area. Make sure to verify which axis is semi-m minor based on the measurement comparison.

Ellipse Formula

To find the area of an ellipse, you can use the straightforward formula: \(A = \pi ab\). Here, \(a\) stands for the semi-major axis length, and \(b\) is the semi-minor axis length. By multiplying these lengths by \(\pi\), you calculate how much space the ellipse covers.

### Application of the Formula

• Ensure you have correctly identified \(a\) and \(b\) based on their values from the standard form equation.
• Multiply \(a\) by \(b\) and then by \(\pi\) to get the area of the ellipse.
• This computation gives a direct measure of the size of the ellipse’s surface area enclosed within its borders.

This formula is a practical tool for solving real-life problems involving elliptical shapes in fields like engineering and architecture.