Certainly! Let's solve the given equation step by step:
The given equation is:
[tex]\[ 49 x^2 + 16 y^2 = 784 \][/tex]
### Step 1: Rewrite the Equation in Standard Form
First, divide both sides by 784 to convert the equation into its standard form for an ellipse:
[tex]\[ \frac{49 x^2}{784} + \frac{16 y^2}{784} = 1 \][/tex]
Simplify the fractions:
[tex]\[ \frac{x^2}{\frac{784}{49}} + \frac{y^2}{\frac{784}{16}} = 1 \][/tex]
[tex]\[ \frac{x^2}{16} + \frac{y^2}{49} = 1 \][/tex]
### Step 2: Identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]
This is now in the standard form for an ellipse:
[tex]\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \][/tex]
where [tex]\(a^2 = 16\)[/tex] and [tex]\(b^2 = 49\)[/tex].
### Step 3: Find the Major and Minor Axes
In this case:
- [tex]\(a = \sqrt{16} = 4\)[/tex]
- [tex]\(b = \sqrt{49} = 7\)[/tex]
Since [tex]\(b > a\)[/tex], the major axis is vertical (along the y-axis) and the minor axis is horizontal (along the x-axis).
### Step 4: Determine the Ends of the Major Axis
The ends of the major axis are at:
[tex]\[ (0, \pm b) = (0, \pm 7) \][/tex]
### Step 5: Find the Foci
The distance of the foci from the center is given by [tex]\(c\)[/tex], where:
[tex]\[ c^2 = b^2 - a^2 \][/tex]
Substitute [tex]\(b^2\)[/tex] and [tex]\(a^2\)[/tex]:
[tex]\[ c^2 = 49 - 16 = 33 \][/tex]
[tex]\[ c = \sqrt{33} \][/tex]
### Final Answer:
- The ends of the major axis are at [tex]\((0, \pm 7)\)[/tex].
- The foci are located at [tex]\((0, \pm \sqrt{33})\)[/tex].
So, in summary:
[tex]\[ \text{Major Axis: } (0, \pm 7) \][/tex]
[tex]\[ \text{Foci: } (0, \pm \sqrt{33}) \][/tex]
