Respuesta :
Answer:
Step-by-step explanation:
You don't have anything listed to choose from but that's ok, we can find them anyways. An ellipse wraps itself around the foci and is either a horizontal or a vertical ellipse. If it is a horizontal ellipse, the foci will share the same y coordinate as the center; if it is a vertical ellipse, the foci will share the same x coordinate as the center. We need to first determine whether this is a horizontal or a vertical ellipse.
In the equation for an ellipse, whether it is horizontal or vetical, there are a and b values. In an ellipse, the a value is ALWAYS bigger than the b value. Because of this, if the a value is under the x coordinate of the ellipse, then the ellipse is horizontal; if the a value is under the y coordinate of the ellipse, then the ellipse is vertical. Because the larger value of a (a^2 = 16 so a = 4) is under the y coordinate of the ellipse, this is a vertical ellipse and the foci will share the same x coordinate as the center. Let's find the center. Our center is (7, -3).
The formula to find the foci (7, -3+c) and (7, -3-c)
[tex]c^2=a^2-b^2[/tex], similar to Pythagorean's theorem. Don't make that mistake. Pythagorean's Theorem is used to find the c value in a hyperbola, not an ellipse. For us,
[tex]c^2=16-4[/tex] so
[tex]c=\sqrt{12}=2\sqrt{3}[/tex]
This means that the foci are located at (7, -3+2√3) and (7, -3-2√3)
Not sure what form you need that in, but the decimal equivalencies are:
(7, .4641016151) and (7, -6.464101615)
Using the equation of the ellipse, it is found that the coordinates of the foci are:
- [tex](7, -3 - \sqrt{12})[/tex] and [tex](7, -3 + \sqrt{12})[/tex].
Equation of an ellipse:
- The equation of a vertical ellipse of center [tex](x_0,y_0)[/tex] is given by:
[tex]\frac{(x - x_0)^2}{b^2} + \frac{(y - y_0)^2}{a^2} = 1[/tex]
- The coordinates of the foci are as follows: [tex](x_0, y_0 \pm c)[/tex], in which the coefficient c is given by:
[tex]c^2 = a^2 - b^2[/tex]
In this problem, the equation is:
[tex]\frac{(x - 7)^2}{4} + \frac{(y + 3)^2}{16} = 1[/tex]
Hence, [tex]x_0 = 7, y_0 = -3, a^2 = 16, b^2 = 4[/tex].
The coefficient c is given by:
[tex]c^2 = a^2 - b^2[/tex]
[tex]c^2 = 16 - 4[/tex]
[tex]c^2 = 12[/tex]
[tex]c = \sqrt{12}[/tex]
The, the foci are located at: [tex](7, -3 - \sqrt{12})[/tex] and [tex](7, -3 + \sqrt{12})[/tex].
To learn more about equation of ellipses, you can take a look at https://brainly.com/question/19507943
