Problem 58 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given characteristics. focus \((2,0), x\) -intercept 4

Step-by-Step Solution

Answer

The standard form equation of the ellipse with the provided characteristics is \( x^2/16 + y^2/12 = 1 \)

Step 1: Identify Given Characteristics

From the problem, it is known that \[ c = 2 \] (the distance from the center to either focus) and \[ a = 4 \] (x-intercept, which is the distance from the center to the end of the major axis).

Step 2: Apply the Formula

Use the formula \[ c^2 = a^2 - b^2 \] to find b (the distance from the center to the end of the minor axis). In this case, \[ b^2 = a^2 - c^2 = 4^2 - 2^2 = 12 \].

Step 3: Write the Equation

Use the values of a and b in the standard form equation of an ellipse: \[ x^2/a^2 + y^2/b^2 = 1 \] to write the equation of the ellipse. The equation is \[ x^2/4^2 + y^2/12 = 1 \]. On Simplifying, we get \[ x^2/16 + y^2/12 = 1 \].

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane at different angles. These sections form various shapes, such as circles, ellipses, parabolas, and hyperbolas. Each shape has unique characteristics based on how the plane intersects the cone. Ellipses stand out due to their oval shape, and they occur when the plane cuts through the cone at an angle that's not parallel to the base.

Ellipses are symmetrical and have two foci, points within the ellipse that help define its shape. The sum of the distances from any point on the ellipse to the two foci is constant.

• Different conic sections can be described using general quadratic equations, but each has a specific standard form.
• Understanding these different forms and properties allows us to describe and analyze the sections mathematically.

Standard Form

The standard form of an ellipse equation helps describe its shape and orientation mathematically, making it easier to analyze and graph.

The standard form of an ellipse with the center at the origin is given by:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where:

• \(a\) is the semi-major axis, the longest radius of the ellipse.
• \(b\) is the semi-minor axis, the shortest radius of the ellipse.

Using this equation, one can seamlessly determine the ellipse's axes lengths and easily plot it on a coordinate plane.

Center at the Origin

The center of an ellipse is the midpoint between the two foci and serves as the symmetry point for the ellipse. When the center is placed at the origin, meaning the point \((0,0)\) on the Cartesian plane, it simplifies the equation and analysis.

With the center at the origin:

• The equation becomes simplified as it merely involves powers of \(x\) and \(y\) without additional terms.
• Calculating variables such as the axes' lengths becomes straightforward.

Placing the center at the origin aids in creating a standardized way to study ellipses and further helps in visualizing its characteristics on a graph.

Major and Minor Axes

The major and minor axes of an ellipse play a significant role in defining its overall shape and size.

• The major axis is the longer diameter that passes through both foci. It represents the greatest width of the ellipse, and its length is determined by \(2a\).
• The minor axis is the shortest diameter that is perpendicular to the major axis and passes through the center of the ellipse. Its length is indicated by \(2b\).

In our specific equation, given data tells us:

• \(a = 4\) as the x-intercept, indicating the semi-major axis length.
• \(b\) can be calculated using the relationship \(c^2 = a^2 - b^2\) where \(c\) is the distance from the center to a focus.

Understanding these axes helps graph the ellipse accurately and comprehend its alignment and dimensions on the plane.