2. What are the foci of the ellipse? Graph the ellipse. (1 point) 18x ^ 2 + 36y ^ 2 = 648
AAAaannndd the bot gets it wrong yet again!
since the major axis is horizontal, the foci lie on the x-axis, not the y-axis
c^2 = 36-18 = 18, so c = 3√2
The foci are (±3√2,0)
furthermore, since a^2=36, a=6
Apologies for the mistake in the previous answer. You are correct that the major axis is horizontal and the foci lie on the x-axis. The correct foci are (±3√2, 0) and the value of a is 6. Thank you for pointing out the error.
To find the foci of an ellipse, we first need to convert the equation of the ellipse into standard form, which is (x-h)^2/a^2 + (y-k)^2/b^2 = 1.
Let's start by factoring out the common factor of 36 from the equation 18x^2 + 36y^2 = 648. We get:
36(0.5x^2 + y^2) = 648
Next, divide both sides of the equation by 648 to get the equation in standard form:
(0.5x^2)/18 + (y^2)/18 = 1
Simplifying further, we have:
(x^2)/36 + (y^2)/18 = 1
Comparing this equation to the standard form, we see that a^2 = 36 and b^2 = 18.
The value of a gives the distance from the center of the ellipse to the ends of the major axis, and the value of b gives the distance from the center to the ends of the minor axis.
To find the foci, we use the formula c^2 = a^2 - b^2, where c is the distance from the center to each focus.
Substituting the values of a^2 and b^2 into the formula, we have:
c^2 = 36 - 18
= 18
Taking the square root of both sides, we find that c = √18 or approximately 4.2426.
Therefore, the foci of the ellipse are located at a distance of approximately 4.2426 units on either side of the center along the major axis.
To graph the ellipse, plot the center point at (0, 0), then plot points that are a distance of 6 units in each direction along the x-axis (since a = 6), and a distance of approximately 3 units in each direction along the y-axis (since b = √18 ≈ 4.2426). Connect these points to form the ellipse shape. The foci should be located on the major axis, 4.2426 units away from the center on each side.
The foci of the ellipse are (0, ±3√2).
To graph the ellipse, first divide both sides of the equation by 648 to get:
x^2/36 + y^2/18 = 1
This gives us the equation in standard form, where a^2 = 36 and b^2 = 18.
We can then find the center of the ellipse by setting both the x and y terms to zero:
x^2/36 + y^2/18 = 0
x^2/36 = -y^2/18
This means the center is at (0, 0).
Next, we can find the major and minor axes by taking the square roots of a^2 and b^2. In this case, a = 6√2 and b = 3√2. The major axis is the horizontal line passing through the center, and the minor axis is the vertical line passing through the center.
To plot the ellipse, we can start by drawing a rectangle whose sides are the major and minor axes. The width of the rectangle is 2a = 12√2 and the height is 2b = 6√2. The center of the ellipse is at the midpoint of the rectangle, which is (0, 0). We can then draw the ellipse to fit inside the rectangle, and the foci are located along the major axis of the ellipse, at (0, ±3√2).
Here is a rough sketch of the ellipse:
```
|
| | |
| o |
| | | |
|
```
The "o" represents the center of the ellipse, and the "|" and "-" symbols represent the major and minor axes respectively. The foci are located at the top and bottom of the ellipse along the major axis.