Hi! I really need help please. I'm really desperate to pass our calculus but I'm having a hard time. If it is okay, can someone answer this question pls.
Find the coordinates of the center, vertices, foci, and extremities of the minor axis, the eccentricity and the equations of the directrices of the ellipse. 🙏
first, the equation. This really should have been the easy part.
20x^2+12y^2-20x+12y=52
let's divide through by 4.
5x^2-5x + 3y^2+3y = 13
5(x^2 - x + 1/4) + 3(y^2 + y + 1/4) = 13 + 5/4 + 3/4
5(x - 1/2)^2 + 3(y - 1/2)^2 = 15
(x - 1/2)^2/3 + (y - 1/2)^2/5 = 1
so now we know that
a^2 = 5
b^2 = 3
c^2 = 2
and the major axis is vertical.
See what you can do with that. Confirm your results at
https://www.wolframalpha.com/input/?i=ellipse+20x%5E2%2B12y%5E2-20x%2B12y%3D52
dividing by 4 ... 5x² + 3y² - 5x + 3y = 13
rearranging ... 5(x² - x) + 3(y² + y) = 13
completing the squares ... 5(x - 1/2)^2 - 5/4 + 3(y + 1/2)^2 - 3/4 = 13
... 5(x - 1/2)^2 + 3(y + 1/2)^2 = 13 + 5/4 + 3/4 = 15
dividing by 15 ... [(x - 1/2)^2 / 3] + [(y + 1/2)^2 / 5] = 1
this is the equation of an ellipse centered at ... (1/2,-1/2)
the semi-major axis is vertical and equal to ... √5
the semi-minor axis is equal to ... √3
Having 20x²+12y-20x+12y=52
Sorry. Typo.
This is the final -> 20x²+12y²-20x+12y=52
Thank u. 😭🙏
Thank u also, R_scott!
Of course, I'd be happy to help you with your calculus question! Solving for the center, vertices, foci, extremities of the minor axis, eccentricity, and equations of the directrices of an ellipse involves the use of some key formulas and concepts. I'll guide you through the process step by step so that you can understand the solution.
To find the coordinates of the center, vertices, and foci of an ellipse, we need to know its equation. The standard equation for an ellipse with center at (h, k) is:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
Where 'a' is the distance from the center to a vertex along the major axis, and 'b' is the distance from the center to a vertex along the minor axis.
Step 1: Identify Center and Axes
First, we need to identify the given information in the problem to determine the values of 'h' and 'k' (the coordinates of the center) and which axis is major and which is minor.
Step 2: Find the Values of 'a' and 'b'
To find the values of 'a' and 'b', we need additional information. This might be given in the problem, or you might need to calculate it using other provided points or equations related to the ellipse.
Step 3: Determine the Coordinates of the Vertices
The vertices of an ellipse are points where the ellipse intersects the major and minor axes. For example, the vertices along the major axis are (h ± a, k), and the vertices along the minor axis are (h, k ± b).
Step 4: Calculate the Eccentricity
The eccentricity of an ellipse is a measure of how "squashed" or elongated it is. It can be calculated using the formula:
e = √(1 - (b^2 / a^2))
Where 'e' represents the eccentricity, 'a' represents the distance from the center to a vertex along the major axis, and 'b' represents the distance from the center to a vertex along the minor axis.
Step 5: Determine the Coordinates of the Foci
To find the coordinates of the foci, we can use the fact that the distance from the center to each focus is given by:
c = √(a^2 - b^2)
Where 'c' represents the distance from the center to a focus. The coordinates of the foci will then be (h ± c, k).
Step 6: Find the Equations of the Directrices
The directrices are lines that are used to define the geometry of the ellipse. The equations of the directrices can be determined using the following formulas:
For an ellipse with a horizontal major axis:
y = k ± (a^2 / c)
For an ellipse with a vertical major axis:
x = h ± (a^2 / c)
Once you have obtained the values for the center, vertices, foci, extremities of the minor axis, eccentricity, and equations of the directrices, you can substitute them into the appropriate formulas to find the solution.
It is important to note that the specific values of the center, vertices, foci, and directrices of the ellipse will depend on the given information of the problem.
I hope this explanation helps! If you have additional questions or need further assistance, please let me know.