Problem 35 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Endpoints of major axis: \((7,9)\) and \((7,3)\) Endpoints of minor axis: \((5,6)\) and \((9,6)\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is given by \(\frac{(x-7)^2}{9} + \frac{(y-6)^2}{4} = 1\).

Step 1: Find the Center of the Ellipse

The center of the ellipse, denoted \((h, k)\), is the midpoint of either the major or minor axis. Add the x-coordinates of the endpoints of any axis and divide by 2 to get the x-coordinate of the center. Similarly, add the y-coordinates of the endpoints of the axis and divide by 2 to get the y-coordinate of the center. So, using the major axis endpoints \((7,9)\) and \((7,3)\), we find \(h = \frac{7 + 7}{2} = 7\) and \(k = \frac{9 + 3}{2} = 6\). So, the center is \((7, 6)\).

Step 2: Determine the Lengths of the Major and Minor Axes

The length of the major axis is the distance between the two points \((h, y1)\) and \((h, y2)\) where \(y1\) and \(y2\) are y-coordinates of the endpoints of the major axis. Similarly, the length of the minor axis is the distance between the two points \((x1, k)\) and \((x2, k)\) where \(x1\) and \(x2\) are x-coordinates of the endpoints of the minor axis. So, major axis length = \(|9 - 3| = 6\) and minor axis length = \(|9 - 5| = 4\). We can get the values of \(a\) and \(b\) by dividing the lengths of the major and minor axes by 2, respectively. So, \(a = 3\) and \(b = 2\).

Step 3: Write the Standard Form of the Equation

Using \(h = 7\), \(k = 6\), \(a = 3\) and \(b = 2\), we write the equation in standard form as \[\frac{(x-7)^2}{3^2} + \frac{(y-6)^2}{2^2} = 1\].