The given equation is:

$\frac{{(x - 3.5)^2}}{2} + \frac{{(y + 13.5)^2}}{5} = 1$

This is the equation of an ellipse in standard form. Let's break it down:

General Form of an Ellipse Equation:

$\frac{{(x - h)^2}}{a^2} + \frac{{(y - k)^2}}{b^2} = 1$

• $(h, k)$ is the center of the ellipse.
• $a$ and $b$ are the semi-major and semi-minor axes, respectively.
• If $a^2 > b^2$, the major axis is along the $x$-axis; if $b^2 > a^2$, the major axis is along the $y$-axis.

Interpretation of the Given Equation:

• $(x - 3.5)$ means the center along the $x$-axis is at $x = 3.5$.
• $(y + 13.5)$ means the center along the $y$-axis is at $y = -13.5$ (since it's written as $(y - (-13.5))$).
• The denominators $2$ and $5$ correspond to $a^2 = 2$ and $b^2 = 5$, so:
  • $a = \sqrt{2} \approx 1.414$
  • $b = \sqrt{5} \approx 2.236$

Characteristics of the Ellipse:

1. Center: $(3.5, -13.5)$
2. Semi-major axis: $\sqrt{5} \approx 2.236$ (since $b^2 > a^2$, the major axis is along the $y$-axis).
3. Semi-minor axis: $\sqrt{2} \approx 1.414$
4. Orientation: Vertical, as the larger denominator (5) is under the $y$-term.

Would you like more details on ellipses or related concepts?

Here are 5 relative questions:

1. How do you find the foci of an ellipse?
2. Can you explain the relationship between the semi-major and semi-minor axes?
3. How would the equation change if the ellipse were centered at the origin?
4. How do you compute the eccentricity of an ellipse?
5. What happens if both denominators are equal in the ellipse equation?

Tip: To identify an ellipse's major axis direction, always compare the denominators under $x$ and $y$—the larger one indicates the major axis direction.