Q.Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse $9x^2 + 4y^2 = 36$.

The key idea is to rewrite the given ellipse in standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then identify $a$ and $b$ (with $b > a$ here since the denominator under $y^2$ is larger, meaning the major axis is vertical).

Step 1: Divide both sides by 36:

Step 2: Here $a^2 = 4$ (under $x^2$) and $b^2 = 9$ (under $y^2$). Since $b > a$, the major axis is along the $y$-axis. So $a = 2$, $b = 3$.

Step 3: For an ellipse, $c^2 = b^2 - a^2 = 9 - 4 = 5$, so $c = \sqrt{5}$. Eccentricity $e = \frac{c}{b} = \frac{\sqrt{5}}{3}$.

Vertices are at $(0, \pm b) = (0, \pm 3)$. Foci are at $(0, \pm c) = (0, \pm \sqrt{5})$. Length of major axis $= 2b = 6$, length of minor axis $= 2a = 4$.

Concept First: Why This Approach Works

An ellipse is the set of points where the sum of distances to two fixed points (foci) is constant. The standard form tells us everything about its shape and orientation.

For an ellipse centered at the origin:

• If the larger denominator is under $x^2$, the ellipse is horizontal (major axis along x-axis).
• If the larger denominator is under $y^2$, the ellipse is vertical (major axis along y-axis).

The given equation is $9x^2 + 4y^2 = 36$. To read off the axes, we must first rewrite it in the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (or the swapped version). The numbers $a$ and $b$ are the semi-major and semi-minor axes — but which is which depends on which is larger.

Step-by-Step Solution

1. Rewrite in standard form

Divide both sides of $9x^2 + 4y^2 = 36$ by $36$:

So $a^2 = 9$ and $b^2 = 4$ (or vice versa — we decide next).

2. Identify which is $a$ (semi-major) and which is $b$ (semi-minor)

Since $9 > 4$, the larger denominator is under $y^2$. That means the major axis is vertical. So:

3. Find $c$ (distance from center to each focus)

For any ellipse, $c^2 = a^2 - b^2$ (where $a$ is the semi-major axis). So:

4. Write the coordinates

Since the major axis is vertical, the foci and vertices lie on the y-axis.

• Vertices: at $(0, \pm a) = (0, \pm 3)$
• Foci: at $(0, \pm c) = (0, \pm \sqrt{5})$

5. Lengths of axes

• Major axis length = $2a = 2 \times 3 = 6$
• Minor axis length = $2b = 2 \times 2 = 4$

6. Eccentricity

Eccentricity $e = \frac{c}{a} = \frac{\sqrt{5}}{3}$. This is a number between 0 and 1, as it should be for an ellipse. The closer to 0, the more circular; here it's about 0.745, so noticeably elongated.