Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,±8)\(;\) foci: (0,±4)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \[ \frac{x^2}{16} + \frac{y^2}{64} = 1 \] .

Step 1: Determine the lengths of the semi-major and semi-minor axes

The vertices of the ellipse are given as (0,±8), so the semi-major axis length is 8. The foci of the ellipse are given as (0,±4), so the distance between the center and the foci, which is the semi-minor axis, is 4. Therefore, \( a = 8 \) and \( b = 4 \).

Step 2: Identify the orientation

Since the vertices and foci are aligned along the y-axis, the ellipse is vertically oriented. This means that \( a \), the length of the semi-major axis, is under the y term in the equation, and \( b \), the length of the semi-minor axis, is under the x term.

Step 3: Write the equation of the ellipse

Substitute the values of a and b into the standard equation of the ellipse. Therefore, the equation of the ellipse is \[ \frac{x^2}{4^2} + \frac{y^2}{8^2} = 1 \] , which simplifies to \[ \frac{x^2}{16} + \frac{y^2}{64} = 1 \] .