To find the vertices of the ellipse given by the equation:

$\frac{x^2}{81} + \frac{y^2}{9} = 1$

Step 1: Identify the values of $a^2$ and $b^2$

The standard form of the ellipse is:

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

For the given ellipse:

• $a^2 = 81$, so $a = \sqrt{81} = 9$
• $b^2 = 9$, so $b = \sqrt{9} = 3$
• The center of the ellipse is at $(h, k) = (0, 0)$

Step 2: Determine the orientation of the ellipse

Since $a^2 = 81 > b^2 = 9$, the major axis is along the x-axis, and the minor axis is along the y-axis.

Step 3: Find the vertices

• Vertices along the major axis (x-axis):

  • The vertices are located $\pm a$ units from the center along the x-axis.
  • Thus, the vertices are at $(h \pm a, k) = (0 \pm 9, 0) = (9, 0)$ and $(-9, 0)$.

• Vertices along the minor axis (y-axis):

  • The vertices are located $\pm b$ units from the center along the y-axis.
  • Thus, the vertices are at $(h, k \pm b) = (0, 0 \pm 3) = (0, 3)$ and $(0, -3)$.

However, since the major axis is along the x-axis, the vertices of the ellipse are:

$\text{Vertices: } (9, 0) \text{ and } (-9, 0)$

Would you like further details or have any questions?

Related Questions:

1. How do you find the foci of an ellipse?
2. What is the relationship between the eccentricity and the shape of an ellipse?
3. How do you find the length of the major and minor axes of an ellipse?
4. How do you derive the equation of an ellipse from the distance formula?
5. What is the significance of the directrix in the context of an ellipse?

Tip:

Remember that the vertices of an ellipse are always located along the major axis. The distance between the center and each vertex is equal to the semi-major axis length ($a$).