Respuesta :

Answer:

It is choice A.

Step-by-step explanation:

The general form is (x - h)^2 / a^2 + (x - k)^2/b^2 = 1 where (h, k) is the center, 2a = major axis and 2b = minor axis.

The ellipse in the question has a^2 > b^2 so the major axis is  parallel to the x axis.

The minor axis  which is parallel to the y-axis is of length 10 so b^2 = (1/2 * 10)^2

= 25 so we can eliminate C.

The center of the ellipse = the midpoint of a line joining the focii so it is:

( 3+ 7)/2,  6)

= (5,6).

As (h, k) is the center we have h = 5 and k = 6.

So it is choice A.

The equation of the eclipse will be [tex]\frac{(x-5)^2}{29} +\frac{(y-6)^2}{25} =1[/tex]  i.e. option A.

What is equation of the ellipse?

The standard equation of the ellipse is [tex]\frac{(x-h)^2}{a^2} +\frac{(y-k)^2}{b^2} =1[/tex].

Here,  

(h, k) is the center, and 2a and 2b are major and minor axis.

We have,

Length of minor axis = 10

i.e. 2b = 10

And,  b = 5

And,

Foci (c) located at (3,6) and (7,6),

i.e. Major axis is parallel to x-axis.   [Because y is constant]

And,

The foci (c) always lie on the major axis.

And,

c² = a² - b²

Now,

The center of an ellipse is the midpoint of both the major and minor axes, i.e.  the midpoint of a line joining the foci (c),

i.e. Center [tex]=( \frac{x_1 + x_2 }{2}, \frac{y_1 + y_2}{2}) =( \frac{3+7}{2}, \frac{6+6}{2}) =(5, 6)[/tex]

Now,

Foci (c) = 5 - 3 = 2

So,

c² = a² - b²

i.e.

2² = a² - 5²

4 =  a² - 25

⇒  a² = 29  

And, b² = 5² = 25

So,

h = 5 and k = 6

Now,

Putting the values in the standard form of equation of the ellipse,

i.e.

[tex]\frac{(x-h)^2}{a^2} +\frac{(y-k)^2}{b^2} =1[/tex]

i.e.

[tex]\frac{(x-5)^2}{29} +\frac{(y-6)^2}{25} =1[/tex]

So, this is the equation of the eclipse i.e. option A.

Hence, we can say that the equation of the eclipse will be [tex]\frac{(x-5)^2}{29} +\frac{(y-6)^2}{25} =1[/tex]  i.e. option A.

To know more about equation of the eclipse click here

https://brainly.com/question/14281133

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