Q.Find the equation for the ellipse that satisfies the given conditions: Ends of major axis $(\pm 3, 0)$, ends of minor axis $(0, \pm 2)$.

Concept: Locus of Point — The ellipse is defined by its major and minor axes lengths, centered at the origin.

Step 1: The ends of the major axis are $(\pm 3, 0)$, so the semi-major axis length is $a = 3$ and the major axis lies along the $x$-axis.

Step 2: The ends of the minor axis are $(0, \pm 2)$, so the semi-minor axis length is $b = 2$ and the minor axis lies along the $y$-axis. …

Why this approach works

The key to writing the equation of an ellipse is identifying its center and the lengths of its axes. The ends of the major and minor axes are the farthest and nearest points on the ellipse from the center, respectively. When the axes are aligned with the coordinate axes (which they are here, since the given points lie on the x‑ and y‑axes), the ellipse has a standard form.

The ends of the major axis $(\pm 3, 0)$ tell us two things at once: the center is at the midpoint, which is $(0,0)$, and the semi‑major axis length is $a = 3$. Similarly, the ends of the minor axis $(0, \pm 2)$ give the semi‑minor axis length $b = 2$. Since the major axis lies along the x‑axis (the $x$‑coordinates change, $y$ stays 0), the standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

Step‑by‑step solution

1. Identify the center.

   The ends of the major axis are $(-3,0)$ and $(3,0)$. Their midpoint is $\left( \frac{-3+3}{2}, \frac{0+0}{2} \right) = (0,0)$.

   The ends of the minor axis are $(0,-2)$ and $(0,2)$, whose midpoint is also $(0,0)$.

   So the ellipse is centered at the origin.

2. Determine the orientation.

   The major axis ends have $y = 0$ and varying $x$ — this means the major axis is horizontal (along the x‑axis). …