Question

1. which graph represents the equation \\(\frac{x^2}{49} + \frac{y^2}{81} = 1\\)

Explanation:

Step1: Identify ellipse standard form

The given equation $\frac{x^2}{49}+\frac{y^2}{81}=1$ matches the vertical major axis ellipse standard form $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ where $a > b$.

Step2: Calculate intercepts

For x-intercepts: Set $y=0$, solve $\frac{x^2}{49}=1 \implies x=\pm7$.
For y-intercepts: Set $x=0$, solve $\frac{y^2}{81}=1 \implies y=\pm9$.

Step3: Match to graph

The middle graph has intercepts $(0, \pm9)$ and $(\pm7, 0)$, which matches the calculated intercepts for the ellipse equation.

Answer:

The middle graph (the full ellipse with intercepts $(0, 7)$, $(0, -7)$, $(9, 0)$, $(-9, 0)$)