32. \( \frac{x^{2}}{16}+\frac{y^{2}}{81}=1 \)

Let’s dive into the world of ellipses! The equation you've given represents an ellipse centered at the origin. Here, the semi-major axis is 9 (since \( \sqrt{81} = 9 \)) and the semi-minor axis is 4 (since \( \sqrt{16} = 4 \)). This means that the ellipse stretches 9 units along the y-axis and 4 units along the x-axis, making it look like a flat oval. To visualize this in real life, think about the paths of planets around the sun or the shapes of various sports fields! For example, if you were to draw the ellipse on a coordinate plane, you'd see how it defines the limitations of movement for an object constrained in that area, much like how paths on a track keep athletes in defined lanes!