Calculating the Area of an Ellipse

The question asks for the area enclosed by the curve given by the equation \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \).

This equation is the standard form of an ellipse centered at the origin.

Understanding the Standard Ellipse Equation

The standard equation for an ellipse centered at the origin is typically written as:

\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Here, \( a^2 \) is the value under the \( x^2 \) term, and \( b^2 \) is the value under the \( y^2 \) term. The values \( a \) and \( b \) represent the lengths of the semi-axes of the ellipse. The semi-major axis is the longer one, and the semi-minor axis is the shorter one.

Relating the Given Equation to the Standard Form

Comparing the given equation \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \) with the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can identify the values of \( a^2 \) and \( b^2 \):

• \( a^2 = 16 \)
• \( b^2 = 25 \)

To find the lengths of the semi-axes, we take the square root:

• \( a = \sqrt{16} = 4 \)
• \( b = \sqrt{25} = 5 \)

In this specific ellipse, the semi-axis along the x-direction has length 4, and the semi-axis along the y-direction has length 5. Since \( 5 > 4 \), the semi-major axis is 5 (along the y-axis), and the semi-minor axis is 4 (along the x-axis).

Area Formula for an Ellipse

The area enclosed by an ellipse with semi-axes of lengths \( a \) and \( b \) is given by the formula:

\( \text{Area} = \pi ab \)

Note that here \( a \) and \( b \) are simply the lengths of the two semi-axes derived from \( a^2 \) and \( b^2 \) in the standard equation, regardless of which is larger or under which variable they appear in the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). The formula always uses the product of the two semi-axis lengths.

Calculating the Area

Using the semi-axes lengths we found, \( a=4 \) and \( b=5 \), we can now calculate the area of the ellipse:

\( \text{Area} = \pi \times a \times b \)

\( \text{Area} = \pi \times 4 \times 5 \)

\( \text{Area} = 20\pi \)

Therefore, the area enclosed by the curve \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \) is \( 20\pi \).

Revision Table: Ellipse Properties

Additional Information: Ellipses and Conic Sections

An ellipse is one of the conic sections, formed by the intersection of a plane and a double-napped cone when the plane is not parallel to the base, the axis, or a generator of the cone. It can also be defined as the locus of all points in a plane such that the sum of the distances from two fixed points (foci) is constant.

• Foci: For the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), if \( b > a \) (major axis along y), the foci are at \( (0, \pm c) \) where \( c^2 = b^2 - a^2 \). If \( a > b \) (major axis along x), the foci are at \( (\pm c, 0) \) where \( c^2 = a^2 - b^2 \). For our ellipse \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( b > a \), so \( c^2 = 25 - 16 = 9 \), and \( c=3 \). The foci are at \( (0, \pm 3) \).
• Eccentricity: The eccentricity (\( e \)) of an ellipse is defined as \( e = c/\text{semi-major axis} \). For an ellipse, \( 0 < e < 1 \). Eccentricity measures how "stretched out" the ellipse is. An eccentricity of 0 corresponds to a circle. For our ellipse, \( e = 3/5 = 0.6 \).
• Circumference: Calculating the exact circumference of an ellipse is more complex than the area and usually requires an elliptic integral. There are approximations, but no simple formula like \( 2\pi r \) for a circle.

Understanding the standard form and key properties like semi-axes and area is fundamental for working with ellipses.