Find an equation for the ellipse.

center:(3,2) a=3c Foci:(1,2)(5,2)

h=3 k=2

I do not know where to start.

c is the distance from the centre to a focal point,

form (3,2) to (5,2) is 2
so c = 2, then a = 6
and we know in the ellipse,
c^2 + b^2 = a^2
4 + b^2 = 36
b^2 = 32

so
(x-3)^2 /36 + (y-2)^2/32 = 1

To find the equation of an ellipse, you can use the standard form equation:

(x-h)^2/a^2 + (y-k)^2/b^2 = 1

First, let's find the value of "b" using the given information.

We are told that the center of the ellipse is (h,k) = (3,2), and the distance from the center to each focus is "c". In this case, "c" is given as 3.

The distance between the foci is 2c. From the given information, we know that the foci are located at (1,2) and (5,2), which means the distance between them is:

2c = 5 - 1 = 4.

Since c = 3, we can find the value of "b" using the equation:

b^2 = c^2 - a^2,
b^2 = 3^2 - c^2 = 3^2 - 3^2 = 9 - 9 = 0.

Now that we have the values of "a" and "b", we can write the equation of the ellipse:

(x-3)^2/3^2 + (y-2)^2/0^2 = 1.

Simplifying this equation, we get:

(x-3)^2/9 + (y-2)^2/0 = 1.

The equation of the ellipse is (x-3)^2/9 + (y-2)^2/0 = 1.

To find the equation for the ellipse, we can use the formula:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

Where (h, k) represents the coordinates of the center of the ellipse and a and b represent the lengths of the major and minor axes, respectively.

From the given information, we know that the center is (3, 2). Now, we need to find the values of a and b.

The distance between the center and one of the foci is denoted by c. We can determine c by finding the distance between the center and one of the foci using the distance formula:

c = √((xf - x)^2 + (yf - y)^2)

Using the coordinates of one of the foci (1, 2) and the center (3, 2):

c = √((1 - 3)^2 + (2 - 2)^2)
c = √((-2)^2 + 0^2)
c = √4
c = 2

Now, we have the value of c, which means we can determine a using the relationship: a = 3c. Therefore, a = 3 * 2 = 6.

To find b, we can use the relationship: a^2 = b^2 + c^2.

Plugging in the values of a and c:

6^2 = b^2 + 2^2
36 = b^2 + 4
b^2 = 36 - 4
b^2 = 32

Now that we have the values of a and b, we can substitute them into the equation formula:

(x - 3)^2 / 6^2 + (y - 2)^2 / (√32)^2 = 1

Simplifying:

(x - 3)^2 / 36 + (y - 2)^2 / 32 = 1

Therefore, the equation of the ellipse is:

(x - 3)^2 / 36 + (y - 2)^2 / 32 = 1