2. What are the foci of the ellipse Graph the ellipse (1 point) 15x ^ 2 - 36t ^ 2 = 645
AAAaannndd the bot gets it wrong yet again!
first of all, the equation is for a hyperbola.
15x ^ 2 - 36t ^ 2 = 645
x^2/43 - y^2/17.9 = 1
c^2 = a^2 + b^2 = 60.9
so c = 7.8
The foci are at (±7.8,0)
plus, I suspect there's a typo in the question ...
I apologize for my mistake. Thank you for correcting me and providing the correct solution. The equation is actually for a hyperbola, and the foci are located at (±7.8, 0). I appreciate your feedback and will strive to improve my accuracy in the future.
To find the foci of an ellipse, we first need to identify the equation of the ellipse in a standard form. The standard form equation of an ellipse centered at the origin is given as:
(x^2 / a^2) + (y^2 / b^2) = 1
If the ellipse is centered at (h, k), the equation becomes:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
In the given equation 15x^2 - 36y^2 = 645, we need to isolate the x term and y term. Divide the equation by 645 to simplify it:
(x^2 / (43/5)) - (y^2 / (215/6)) = 1
Comparing this with the standard form equation, we have:
(x - 0)^2 / (sqrt(43/5))^2 - (y - 0)^2 / (sqrt(215/6))^2 = 1
Now, the equation is in standard form, with a = sqrt(43/5) and b = sqrt(215/6). To find the foci, we can use the formula:
c = sqrt(a^2 - b^2)
Plugging in the values, we can calculate:
c = sqrt((sqrt(43/5))^2 - (sqrt(215/6))^2)
Simplifying further, we obtain:
c = sqrt(43/5 - 215/6)
= sqrt((258/30) - (215/30))
= sqrt(43/30)
Thus, the foci of the given ellipse are located at (±sqrt(43/30), 0).
To find the foci of the ellipse, we first need to put the equation in standard form, which is:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
Where (h,k) is the center of the ellipse, 'a' is the distance from the center to the vertices, and 'b' is the distance from the center to the co-vertices.
To get the equation in this form, we need to divide both sides by 645:
15x^2/645 - 36y^2/645 = 1
Simplifying, we get:
x^2/43 - y^2/(43/5) = 1
So, the center is (0,0), 'a' is sqrt(43), and 'b' is sqrt(43/5).
The distance between the center and each focus is c, where c^2 = a^2 - b^2. Plugging in our values for 'a' and 'b', we get:
c^2 = 43 - 43/5
c^2 = (215/5) - 43/5
c^2 = 172/5
c ≈ 5.2
So, the distance between the center and each focus is approximately 5.2. The foci are located at (±c,0), or approximately (5.2,0) and (-5.2,0).
To graph this ellipse, we can plot the center at (0,0), then sketch the ellipse using the values of 'a' and 'b' as our semi-major and semi-minor axes, respectively. We can also plot the foci at (5.2,0) and (-5.2,0).
(Note: The ellipse is elongated along the x-axis, which means 'a' is greater than 'b'.)
Here is a rough sketch of the graph:
