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 \[(x-7)^2/4 + (y-6)^2/9 = 1\].

Step 1: Identify the Center of the Ellipse

The center of the ellipse is the midpoint of both the major and minor axes. For the major axis with endpoints at \((7,9)\) and \((7,3)\), the midway point can be found using the midpoint formula [(x1+x2)/2, (y1+y2)/2] which gives [(7+7)/2, (9+3)/2] resulting in \((7,6)\). This is also the midpoint of the minor axis with endpoints \((5,6)\) and \((9,6)\). Therefore, the center of the ellipse is at \((7,6)\).

Step 2: Identify the Values of a and b

The distance from the center to either endpoint of the major axis is the value of 'a'. The distance from the center to either endpoint of the minor axis is the value of 'b'. The value of 'a' can be calculated by the Euclidean distance formula sqrt[(x2-x1)^2 + (y2-y1)^2] that gives sqrt[(7-7)^2 + (9-6)^2] which equals 3. The value of 'b', calculated in the same way, equals 2.

Step 3: Write the Standard Form of the Equation

With center at coordinates (h,k), and major axis of length 2a and minor axis of length 2b, the standard form of the equation of an ellipse is \[(x-h)^2/b^2 + (y-k)^2/a^2 = 1\]. Substituting h=7, k=6, a=3 and b=2, the equation becomes: \[(x-7)^2/4 + (y-6)^2/9 = 1\].