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