To find the vertices of the ellipse given by the equation [tex]\(4x^2 + 9y^2 = 36\)[/tex], follow these steps:
1. Rewrite the Ellipse Equation in Standard Form:
We start with the given equation:
[tex]\[
4x^2 + 9y^2 = 36
\][/tex]
To convert the equation to the standard form of an ellipse, we divide all terms by 36:
[tex]\[
\frac{4x^2}{36} + \frac{9y^2}{36} = \frac{36}{36}
\][/tex]
Simplifying the fractions, we get:
[tex]\[
\frac{x^2}{9} + \frac{y^2}{4} = 1
\][/tex]
This is now in the standard form of an ellipse [tex]\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)[/tex], where [tex]\(a^2 = 9\)[/tex] and [tex]\(b^2 = 4\)[/tex].
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
From the standard form equation:
[tex]\[
a^2 = 9, \quad \text{so} \quad a = \sqrt{9} = 3
\][/tex]
[tex]\[
b^2 = 4, \quad \text{so} \quad b = \sqrt{4} = 2
\][/tex]
3. Determine the Vertices:
In the standard form [tex]\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)[/tex], the vertices of the ellipse are located at [tex]\((\pm a, 0)\)[/tex] and [tex]\((0, \pm b)\)[/tex].
Therefore, the vertices for our given ellipse are:
[tex]\[
(\pm 3, 0) \quad \text{and} \quad (0, \pm 2)
\][/tex]
4. List All Vertices:
Writing out these points explicitly, the vertices are:
[tex]\[
(3, 0), \quad (-3, 0), \quad (0, 2), \quad (0, -2)
\][/tex]
Thus, the vertices of the ellipse [tex]\(4x^2 + 9y^2 = 36\)[/tex] are [tex]\(\boxed{( \pm 3, 0), (0, \pm 2)}\)[/tex]. The correct choice among the given options is [tex]\(( \pm 3, 0)\)[/tex].
