Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length 8; length of minor axis = 4; center: (0, 0)

Verified step by step guidance

1. Identify the center of the ellipse, which is given as (0, 0). This means the ellipse is centered at the origin.
2. Since the major axis is horizontal with length 8, find the value of \( a \), which is half the length of the major axis: \( a = \frac{8}{2} = 4 \).
3. The length of the minor axis is 4, so find the value of \( b \), which is half the length of the minor axis: \( b = \frac{4}{2} = 2 \).
4. Because the major axis is horizontal, the standard form of the ellipse equation centered at the origin is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Substitute \( a = 4 \) and \( b = 2 \) into the equation.
5. Write the equation explicitly with the substituted values: \[ \frac{x^2}{4^2} + \frac{y^2}{2^2} = 1 \] This is the standard form of the ellipse equation with the given conditions.

Key Concepts

Standard Form of an Ellipse Equation

The standard form of an ellipse equation depends on the orientation of its major axis. For a horizontal major axis centered at (h, k), the equation is ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1, where 'a' is the semi-major axis length and 'b' is the semi-minor axis length.

Major and Minor Axes Lengths

The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. Their lengths are 2a and 2b respectively, where 'a' > 'b'. Knowing these lengths allows you to find 'a' and 'b', which are essential for writing the ellipse equation.

Center of the Ellipse

The center of the ellipse (h, k) is the midpoint of both the major and minor axes. It shifts the ellipse from the origin if not at (0, 0). In this problem, the center is at (0, 0), simplifying the equation by removing horizontal and vertical shifts.