Problem 36 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Endpoints of major axis: \((2,2)\) and \((8,2)\) Endpoints of minor axis: \((5,3)\) and \((5,1)\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \((\frac{(x-5)^2}{9}) + (\frac{(y-2)^2}{1}) = 1\).

Step 1: Determine the Center of the Ellipse

Firstly, locate the center of the ellipse. This point is the midpoint of either the major or minor axis. Since the endpoints of the major axis are (2,2) and (8,2), the midpoint can be computed as \((\frac{1}{2}(2+8), \frac{1}{2}(2+2)) = (5,2)\). Therefore, the center of the ellipse is at (5,2).

Step 2: Determine the Lengths of the Semi-major and Semi-minor Axes

Next, calculate the distances from the center to an endpoint of each axis to determine the lengths of the semi-major and minor axes. The distance from the center (5,2) to an endpoint of the major axis (2,2) or (8,2) is \(\sqrt{(5-2)^2 + (2-2)^2} = 3\). This is the length of the semi-major axis, denoted 'a'. The distance to an endpoint of the minor axis (5,3) or (5,1) is \(\sqrt{(5-5)^2 + (2-3)^2} = 1\). This length is the semi-minor axis, denoted 'b'. Thus, a = 3 and b = 1.

Step 3: Write the Equation of the Ellipse in Standard Form

The standard form of an ellipse with a center at (h,k), semi-major axis 'a' along the x-axis and semi-minor axis 'b' along the y-axis is \((\frac{(x-h)^2}{a^2}) + (\frac{(y-k)^2}{b^2}) = 1\). Substituting h = 5, k = 2, a = 3 and b = 1, we get \((\frac{(x-5)^2}{9}) + (\frac{(y-2)^2}{1}) = 1\).