To find the foci of the ellipse given by the equation [tex]\(\frac{x^2}{9}+\frac{y^2}{25}=1\)[/tex], we follow these steps:
1. Identify the standard form of the ellipse equation:
The equation of an ellipse in standard form is given as:
[tex]\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the semi-major and semi-minor axes, respectively. For an ellipse, [tex]\(a \geq b\)[/tex].
2. Determine [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
From the given equation [tex]\(\frac{x^2}{9}+\frac{y^2}{25}=1\)[/tex]:
[tex]\[
\frac{x^2}{9} + \frac{y^2}{25} = 1
\][/tex]
we see that [tex]\(a^2 = 25\)[/tex] and [tex]\(b^2 = 9\)[/tex].
3. Calculate the focal distance [tex]\(c\)[/tex]:
For an ellipse, the focal distance [tex]\(c\)[/tex] is determined by the equation:
[tex]\[
c^2 = a^2 - b^2
\][/tex]
Plugging in the values for [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
[tex]\[
c^2 = 25 - 9 = 16
\][/tex]
Solving for [tex]\(c\)[/tex]:
[tex]\[
c = \sqrt{16} = 4
\][/tex]
4. Determine the coordinates of the foci:
Since [tex]\(a^2\)[/tex] corresponds to the term with [tex]\(y^2\)[/tex], it indicates that the major axis is along the [tex]\(y\)[/tex]-axis. Therefore, the foci are located along the [tex]\(y\)[/tex]-axis at coordinates [tex]\((0, \pm c)\)[/tex].
Given [tex]\(c = 4\)[/tex], the coordinates of the foci are:
[tex]\[
(0, 4) \text{ and } (0, -4)
\][/tex]
5. Select the correct answer:
The foci of the ellipse are [tex]\((0, \pm 4)\)[/tex].
Based on this calculation, the correct choice is:
[tex]\[
(0, \pm 4)
\][/tex]
