Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (0,2),(8,2)\(;\) minor axis of length 2

Step-by-Step Solution

Answer

The standard form equation of the ellipse is \((x-4)^2/16 + (y-2)^2/1 = 1\).

Step 1: Determine the center of the ellipse

The center of the ellipse is the midpoint of the line segment connecting the vertices. In our case, the vertices are (0,2) and (8,2). Hence, the midpoint can be calculated by averaging the x and y coordinates. The x-coordinate of the midpoint \(C\) is \((0 + 8)/2 = 4\) and the y-coordinate is \((2+2)/2 = 2\). Therefore, the center \(C\) of the ellipse is at (4,2).

Step 2: Determine the lengths of major and minor axes

The given vertices are on major axis and the distance between these vertices indicates the length of the major axis which is 8 units. This implies, 'a', the semi-major axis length is half of 8, i.e. 4 units. The given minor axis length is 2 units, which signifies 'b' is 2/2 = 1.

Step 3: Write down the equation of the ellipse

The standard form of the equation of an ellipse with center at (h,k), semi-major axis of length a, and semi-minor axis of length b, aligned along the x-axis is \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\). Here, we have h=4, k=2 , a=4, and b=1. Substituting these values into the ellipse equation formula, we get: \((x-4)^2/4^2 + (y-2)^2/1^2 = 1\).