Problem 29 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: \((-2,0),(2,0) ; y\) -intercepts: \(-3\) and 3

Step-by-Step Solution

Answer

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

Step 1: Find the Center, a and c

The foci of the ellipse are given as \((-2,0)\) and \((2,0)\). The center, \((h,k)\), is the midpoint of the foci, which in this case is \((0,0)\). The distance from the center to each focus gives \(c=2\). The distance from the center to a vertex is \(a\) and equals the y-intercept, thus \(a=3\).

Step 2: Find the Value of b

Using the formula \(c^{2}=a^{2}-b^{2}\), and substituting \(c=2\) and \(a=3\), we can solve for \(b\). \(b\) is therefore \(\sqrt{a^{2}-c^{2}} = \sqrt{3^{2}-2^{2}}=\sqrt{5}\). Thus, the semi-minor axis, b, is \(\sqrt{5}\).

Step 3: Write the Equation of the Ellipse

The standard form of the equation of this ellipse is \( (x-h)^{2}/a^{2} + (y-k)^{2}/b^{2} = 1 \). Substituting \(h=0, k=0, a=3, b=\sqrt{5}\), we get the equation as \( x^{2}/9 + y^{2}/5 = 1 \).