Find the major vertices of the ellipse \( 5 x^{2}+3 y^{2}-15=0 \) A \( \quad(5,0) \) and \( (-5,0) \) B \( \quad(5,0) \) and \( (0,-5) \) C \( \quad(\sqrt{5}, 0) \) and \( (-\sqrt{5}, 0) \) D \( \quad(0, \sqrt{5}) \) and \( (0,-\sqrt{5}) \)

To find the major vertices of the given ellipse, we first rewrite the equation \( 5x^2 + 3y^2 - 15 = 0 \) in standard form by isolating the constant: \[ 5x^2 + 3y^2 = 15 \] Dividing every term by 15 gives: \[ \frac{x^2}{3} + \frac{y^2}{5} = 1 \] This is the standard form of an ellipse with \( a^2 = 5 \) and \( b^2 = 3 \). Since \( a^2 > b^2 \), the major axis is along the y-axis. Thus, the vertices along the major axis are located at: \[ (0, \sqrt{5}) \quad \text{and} \quad (0, -\sqrt{5}) \] So, the major vertices of the ellipse are \( (0, \sqrt{5}) \) and \( (0, -\sqrt{5}) \). The correct answer is D. To have a deeper understanding, let's explore the significance of the semi-major and semi-minor axes and how they impact the geometry of ellipses. The distance \( a = \sqrt{5} \) represents the distance from the center at (0,0) along the y-axis, while \( b = \sqrt{3} \) represents the distance along the x-axis. This configuration creates a "stretched" appearance of the ellipse along the y-direction! Additionally, you might find it interesting how ellipses are not just confined to pure mathematics, but appear in physics as the paths of planets in motion. In fact, planets orbiting the sun follow elliptical trajectories according to Kepler’s laws! This connection between mathematics and cosmic phenomena highlights the beauty and applicability of mathematical concepts beyond the classroom.