To solve the problem for the given ellipse equation [tex]\(\frac{x^2}{81} + \frac{y^2}{16} = 1\)[/tex], we need to find the ends of the major axis and the foci. Here's a step-by-step solution:
1. Identify Ellipse Parameters:
- The standard form of the equation of an ellipse is [tex]\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)[/tex].
- Compare [tex]\(\frac{x^2}{81} + \frac{y^2}{16} = 1\)[/tex] with the standard form.
- Here, [tex]\(a^2 = 81\)[/tex] and [tex]\(b^2 = 16\)[/tex].
2. Calculate [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a = \sqrt{a^2} = \sqrt{81} = 9\)[/tex].
- [tex]\(b = \sqrt{b^2} = \sqrt{16} = 4\)[/tex].
3. Determine the Major Axis:
- The lengths of the major and minor axes are determined by [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
- Since [tex]\(a > b\)[/tex], the major axis is along the x-axis.
- The ends of the major axis are at [tex]\((±a, 0)\)[/tex].
- So, the ends of the major axis are at [tex]\((±9, 0)\)[/tex].
4. Calculate the Distance to Foci ([tex]\(c\)[/tex]):
- The distance from the center to each focus, [tex]\(c\)[/tex], is found using the relationship [tex]\(c^2 = a^2 - b^2\)[/tex].
- Substitute the known values: [tex]\(c^2 = 81 - 16 = 65\)[/tex].
- Therefore, [tex]\(c = \sqrt{65}\)[/tex].
5. Determine the Foci:
- The foci are located along the major axis at [tex]\((±c, 0)\)[/tex].
- So, the foci are at [tex]\((±\sqrt{65}, 0)\)[/tex].
Combining these results, we have:
- The ends of the major axis are [tex]\( ( \pm 9, 0 ) \)[/tex].
- The foci are [tex]\( ( \pm \sqrt{65}, 0 ) \)[/tex].
