Answer :
To graph the ellipse and locate the foci for the given equation [tex]\( 64x² + 49y² = 3136 \)[/tex], we need to follow a few steps to convert the equation into its standard form and then find the necessary parameters.
### Step-by-Step Solution:
1. Rewrite the equation in standard form:
The given equation of the ellipse is:
[tex]\[ 64x² + 49y² = 3136 \][/tex]
To convert this to the standard form of an ellipse [tex]\(\left(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \right)\)[/tex], divide both sides by 3136:
[tex]\[ \frac{64x²}{3136} + \frac{49y²}{3136} = 1 \][/tex]
Simplifying the fractions:
[tex]\[ \frac{x²}{49} + \frac{y²}{64} = 1 \][/tex]
2. Identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
In the standard form [tex]\(\frac{x²}{a²} + \frac{y²}{b²} = 1\)[/tex],
[tex]\[ a² = 49 \][/tex]
[tex]\[ b² = 64 \][/tex]
3. Calculate [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ a = \sqrt{49} = 7 \][/tex]
[tex]\[ b = \sqrt{64} = 8 \][/tex]
Since [tex]\(b > a\)[/tex], it indicates that the ellipse is vertically oriented.
4. Locate the foci:
To find the distance [tex]\(c\)[/tex] from the center to each focus, use the relationship for ellipses [tex]\( c² = |b² - a²| \)[/tex]:
[tex]\[ c² = |b² - a²| = |64 - 49| = 15 \][/tex]
[tex]\[ c = \sqrt{15} \approx 3.873 \][/tex]
The foci for a vertically oriented ellipse are located at [tex]\((0, \pm c)\)[/tex]:
[tex]\[ \text{Foci}: \left( 0, \pm \sqrt{15} \right) \][/tex]
[tex]\[ \text{Foci}: (0, 3.873) \text{ and } (0, -3.873) \][/tex]
### Graphing the Ellipse:
To graph the ellipse, we need to plot its major and minor axes and the foci.
1. Plot the center:
The center of the ellipse is at the origin [tex]\((0,0)\)[/tex].
2. Plot the vertices:
Since [tex]\(b = 8\)[/tex], the vertices along the major (vertical) axis are:
[tex]\[ (0, 8) \text{ and } (0, -8) \][/tex]
Since [tex]\(a = 7\)[/tex], the vertices along the minor (horizontal) axis are:
[tex]\[ (7, 0) \text{ and } (-7, 0) \][/tex]
3. Plot the foci:
The foci are at:
[tex]\[ (0, 3.873) \text{ and } (0, -3.873) \][/tex]
### Summary of key points:
- Equation in standard form: [tex]\[ \frac{x^2}{49} + \frac{y^2}{64} = 1 \][/tex]
- Vertices on the major axis: [tex]\[ (0, 8) \text{ and } (0, -8) \][/tex]
- Vertices on the minor axis: [tex]\[ (7, 0) \text{ and } (-7, 0) \][/tex]
- Foci: [tex]\[ (0, 3.873) \text{ and } (0, -3.873) \][/tex]
Using all these points, you can now draw the ellipse with a major vertical axis and locate the foci as indicated above.
### Step-by-Step Solution:
1. Rewrite the equation in standard form:
The given equation of the ellipse is:
[tex]\[ 64x² + 49y² = 3136 \][/tex]
To convert this to the standard form of an ellipse [tex]\(\left(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \right)\)[/tex], divide both sides by 3136:
[tex]\[ \frac{64x²}{3136} + \frac{49y²}{3136} = 1 \][/tex]
Simplifying the fractions:
[tex]\[ \frac{x²}{49} + \frac{y²}{64} = 1 \][/tex]
2. Identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
In the standard form [tex]\(\frac{x²}{a²} + \frac{y²}{b²} = 1\)[/tex],
[tex]\[ a² = 49 \][/tex]
[tex]\[ b² = 64 \][/tex]
3. Calculate [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ a = \sqrt{49} = 7 \][/tex]
[tex]\[ b = \sqrt{64} = 8 \][/tex]
Since [tex]\(b > a\)[/tex], it indicates that the ellipse is vertically oriented.
4. Locate the foci:
To find the distance [tex]\(c\)[/tex] from the center to each focus, use the relationship for ellipses [tex]\( c² = |b² - a²| \)[/tex]:
[tex]\[ c² = |b² - a²| = |64 - 49| = 15 \][/tex]
[tex]\[ c = \sqrt{15} \approx 3.873 \][/tex]
The foci for a vertically oriented ellipse are located at [tex]\((0, \pm c)\)[/tex]:
[tex]\[ \text{Foci}: \left( 0, \pm \sqrt{15} \right) \][/tex]
[tex]\[ \text{Foci}: (0, 3.873) \text{ and } (0, -3.873) \][/tex]
### Graphing the Ellipse:
To graph the ellipse, we need to plot its major and minor axes and the foci.
1. Plot the center:
The center of the ellipse is at the origin [tex]\((0,0)\)[/tex].
2. Plot the vertices:
Since [tex]\(b = 8\)[/tex], the vertices along the major (vertical) axis are:
[tex]\[ (0, 8) \text{ and } (0, -8) \][/tex]
Since [tex]\(a = 7\)[/tex], the vertices along the minor (horizontal) axis are:
[tex]\[ (7, 0) \text{ and } (-7, 0) \][/tex]
3. Plot the foci:
The foci are at:
[tex]\[ (0, 3.873) \text{ and } (0, -3.873) \][/tex]
### Summary of key points:
- Equation in standard form: [tex]\[ \frac{x^2}{49} + \frac{y^2}{64} = 1 \][/tex]
- Vertices on the major axis: [tex]\[ (0, 8) \text{ and } (0, -8) \][/tex]
- Vertices on the minor axis: [tex]\[ (7, 0) \text{ and } (-7, 0) \][/tex]
- Foci: [tex]\[ (0, 3.873) \text{ and } (0, -3.873) \][/tex]
Using all these points, you can now draw the ellipse with a major vertical axis and locate the foci as indicated above.
