But geometrically: ellipse \( \frac{x^2}{100} + \frac{y^2}{64} = 1 \) has semi-major axis 10 (x), semi-minor 8 (y).

["Understanding the Ellipse: ( \frac{x^2}{100} + \frac{y^2}{64} = 1 )", "When studying conic sections, the ellipse stands out as a smooth, closed curve defined by specific geometric properties. Let’s explore the ellipse given by the equation:", "[\n\frac{x^2}{100} + \frac{y^2}{64} = 1\n]", "### Geometric Axes: Semi-Major and Semi-Minor Axes", "This standard form reveals two key axis lengths:", "- Semi-major axis (longest radius) = ( a = 10 ) (along the ( x )-axis)\n- Semi-minor axis (shortest radius) = ( b = 8 ) (along the ( y )-axis)", "Since the denominator under ( x^2 ) (which is larger) is larger than that under ( y^2 ), the ellipse stretches wider horizontally. Thus, the ( x )-axis defines the major axis, confirming that the semi-major axis length is 10 units, and the semi-minor axis length is 8 units.", "### Key Visual Insights", "- The eclipse is centered at the origin ((0, 0)).\n- The vertices on the major axis are at ( (10, 0) ) and ( (-10, 0) ).\n- The co-vertices on the minor axis are at ( (0, 8) ) and ( (0, -8) ).\n- The shape looks like a stretched circle, broader in the horizontal direction.", "### Importance in Math and Science", "Understanding these axis lengths helps in modeling orbits in astronomy, designing reflective surfaces in optics, and solving problems in physics involving elliptical motion. Knowing ( a = 10 ) and ( b = 8 ) allows precise sketching, distance calculations, and analysis of elliptical geometry.", "---", "Summary:\nFor the ellipse ( \frac{x^2}{100} + \frac{y^2}{64} = 1 ), the semi-major axis along the ( x )-axis is ( 10 ), and the semi-minor axis along the ( y )-axis is ( 8 ). This geometric insight is foundational for working with ellipses in both theory and application."]