Problem 25 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. $$\text { Foci: }(-5,0),(5,0) ; \text { vertices: }(-8,0),(8,0)$$

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \((\frac{x^2}{64}) + (\frac{y^2}{39}) = 1\).

Step 1: Determine the distance from the center to the vertices (a)

The vertices are the points (-8,0) and (8,0). The vertices lie on the major axis, at a distance of \(a\) from the center, hence \(a = 8\).

Step 2: Determine the distance from the center to the foci (c)

The foci are the points (-5,0) and (5,0). The distance from the center to any foci is \(c\), meaning \(c = 5\).

Step 3: Compute the distance from the center to the co-vertices (b)

We can now compute \(b\) or the distance to co-vertices, using the relationship between \(a, b, c\) for ellipses: \(c^2 = a^2 - b^2 => b^2 = a^2 - c^2 => b^2 = 8^2 - 5^2 => b^2 = 39\).

Step 4: Write down the standard form

Having found both \(a\) and \(b\), the standard form equation of the ellipse is \((\frac{x^2}{a^2}) + (\frac{y^2}{b^2}) = 1 => (\frac{x^2}{64}) + (\frac{y^2}{39}) = 1\).