Problem 28 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. $$\text { Foci: }(0,-3),(0,3) ; \text { vertices: }(0,-4),(0,4)$$

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \( \frac{x^2}{7} + \frac{y^2}{16} = 1 \)

Step 1: Determine values of a, b, and c

For the given ellipse, the distance from the center to a vertex (a) is 4 and the distance from the center to the foci (c) is 3. Hence, \(a=4\) and \(c=3\). The distance \(b\) can be calculated from the relationship between a, b, and c in an ellipse, which is \(a^2 = b^2+c^2\). Substituting a=4 and c=3 into this equation gives \(b^2 = a^2 - c^2 = 4^2 - 3^2 = 7\).

Step 2: Write down the standard form of the ellipse equation

The standard form of an ellipse centered at the origin with vertical major axis is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \). Substituting \(a=4\) and \(b = \sqrt{7}\) gives the equation of the ellipse as \( \frac{x^2}{7} + \frac{y^2}{16} = 1 \)