How to Find the Foci of an Ellipse Using Major and Minor Axes | Precalculus

Calculating Ellipse Foci

• 💡 The process to find the foci of an ellipse involves calculating the focal length using the lengths of the major and minor axes.
• 🔑 The formula for focal length (f) is the square root of the major axis squared minus the minor axis squared.

Determining Axis Lengths

• 🎯 The center of the ellipse is identified at (-2, 5). The major axis extends from (5, -2) to (5, 4.5), giving a length of 6.5 units.
• 📊 The minor axis extends from (5, -2) to (7.5, -2), resulting in a length of 2.5 units.

Calculating Focal Length

• 🔬 Using the formula, f = sqrt(6.5^2 - 2.5^2), the calculation yields a focal length of 6.

Locating the Foci

• 📍 The foci lie on the major axis. Starting from the center (5, -2), move 6 units up along the major axis to find the first focus at (5, 4).
• 📍 Moving 6 units down from the center along the major axis locates the second focus at (5, -8).
• ✅ The identified foci for the ellipse are at points (5, 4) and (5, -8).