2. What are the foci of the ellipse? Graph the ellipse. (1 point) 18x ^ 2 + 361 ^ 2 = 648
To find the foci of the ellipse, we need to first put the equation into standard form and identify the values of a and b:
18x^2 + 361y^2 = 648
Divide by 648 to get:
(x^2/36) + (y^2/648/361) = 1
a^2 = 36, so a = 6
b^2 = 648/361, so b ≈ 1.622
c^2 = a^2 - b^2 = 36 - 648/361 = 36 - 1.794 = 34.206
c ≈ 5.847
The foci are located at (0, ±c), which gives us:
(0, 5.847) and (0, -5.847)
To graph the ellipse, we can plot the center at (0,0), the semi-major axis of length 6 in the x-direction and the semi-minor axis of length √(648/361) ≈ 1.125 in the y-direction, and then sketch the curve. It should look like a horizontally stretched ellipse centered at the origin.
To determine the foci of the ellipse with the equation 18x^2 + 361y^2 = 648, we need to rewrite the equation in standard form:
Divide both sides of the equation by 648 to normalize the equation:
(x^2)/36 + (y^2)/648 = 1
Since the coefficient of x^2 is smaller than the coefficient of y^2, it indicates that the major axis is vertical. Therefore, the formula for the ellipse in standard form can be written as:
(x-h)^2 / a^2 + (y-k)^2 / b^2 = 1
Comparing both equations, we can see that h = 0, k = 0, a^2 = 36, and b^2 = 648.
Taking the square root of a^2 and b^2, we find that a = 6 and b = √648 = 18√2.
The distance c, between the center and the foci, is given by the formula c = √(a^2 - b^2).
Substituting the values, we find c = √(36 - 648) = √-612 = √(-1 * 612).
Since the value under the square root is negative, it indicates that the ellipse is imaginary, and there are no real foci for this ellipse.
Now let's graph the ellipse:
To graph the ellipse, we need to determine the values of x and y that satisfy the equation for different values of the parameter t.
By setting y = 0, we can solve for x:
18x^2 = 648
x^2 = 648/18
x^2 = 36
x = ±6
So, when y = 0, we have two points on the x-axis: (-6, 0) and (6, 0).
Similarly, by setting x = 0, we can solve for y:
(361y^2)/648 = 1
361y^2 = 648
y^2 = 648/361
y^2 ≈ 1.795
y ≈ ±1.340
So, when x = 0, we have two points on the y-axis: (0, -1.340) and (0, 1.340).
Now we can sketch the graph of the ellipse:
(Note: Since the equation provided is currently in the form 18x^2 + 361^2 = 648, it appears to be incorrect. The equation should be revised for accurate analysis and graphing.)
To find the foci of an ellipse, we need the equation of the ellipse in standard form. The given equation of the ellipse is 18x^2 + 361y^2 = 648.
To find the standard form equation of an ellipse, we need to divide both sides of the given equation by the constant term. In this case, divide both sides of the equation by 648:
(18x^2)/648 + (361y^2)/648 = 1
Simplify this equation:
x^2/36 + y^2/(648/361) = 1
The equation is now in the standard form: (x^2/a^2) + (y^2/b^2) = 1, where 'a' is the length of the semi-major axis and 'b' is the length of the semi-minor axis.
In our equation, a^2 = 36, so a = √36 = 6. The length of the semi-major axis is 6.
Also, b^2 = (648/361), so b = √(648/361) = √648/√361 = (18/19)√2. The length of the semi-minor axis is (18/19)√2.
The foci of an ellipse can be calculated using the formula c = √(a^2 - b^2), where c is the distance from the center to each focus point.
In our equation, c = √(6^2 - (18/19)√2)^2) = √(36 - (324/361)(2)) = √(36 - 648/361) = √(361(36 - 648/361))/361 ≈ √(361(32592 - 648))/361 ≈ √(361(31944))/361
Approximating, c ≈ √11526404/361 ≈ 107.78/19 ≈ 5.67.
Thus, the foci of the ellipse are located at a distance of approximately 5.67 units from the center along the major axis.
To graph the ellipse, plot the center point at (0,0), which is the intersection of the x and y axes. Then, go 6 units to the right and left from the center along the x-axis to mark the end points of the major axis.
Now, go (18/19)√2 units up and down from the center along the y-axis to mark the end points of the minor axis.
Using these points, draw the ellipse by smoothly connecting the end points of the major and minor axes.
Finally, draw two points on both sides of the center, approximately 5.67 units away along the major axis. These points represent the foci of the ellipse.
Note: It is important to remember that graphing an ellipse accurately requires precise measurements and scale on the axes.