The given equation is $36x^2 + 4y^2 = 144$.

Step 1: Converting to standard form

Dividing both sides by 144:

$$\frac{36x^2}{144} + \frac{4y^2}{144} = 1$$ $$\frac{x^2}{4} + \frac{y^2}{36} = 1$$

Step 2: Identifying a and b

Since the denominator of $\frac{y^2}{36}$ is larger than the denominator of $\frac{x^2}{4}$, the major axis is along the y-axis.

Comparing with $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$:

• $a^2 = 36$, so $a = 6$
• $b^2 = 4$, so $b = 2$

Step 3: Calculating c and eccentricity

$c = \sqrt{a^2 - b^2} = \sqrt{36 - 4} = \sqrt{32} = 4\sqrt{2}$

Eccentricity $e = \frac{c}{a} = \frac{4\sqrt{2}}{6} = \frac{2\sqrt{2}}{3}$

Step 4: Finding all parameters

• Foci: $(0, \pm 4\sqrt{2})$ i.e., $(0, 4\sqrt{2})$ and $(0, -4\sqrt{2})$
• Vertices: $(0, \pm 6)$ i.e., $(0, 6)$ and $(0, -6)$
• Length of major axis: $2a = 12$ units
• Length of minor axis: $2b = 4$ units
• Eccentricity: $\frac{2\sqrt{2}}{3}$
• Length of latus rectum: $\frac{2b^2}{a} = \frac{2 \times 4}{6} = \frac{4}{3}$ units