Given an ellipse; 8x^2+y^2+80x-6y+193=0, find center, major axis, minor axis, & distance from c to foci?

To find the center, major axis, minor axis, and distance from the center to the foci of an ellipse, we need to rewrite the equation of the ellipse in standard form. The standard form of an ellipse is:

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

Where (h, k) represents the center of the ellipse, 'a' represents half the length of the major axis, 'b' represents half the length of the minor axis, and 'c' represents the distance from the center to the foci.

Let's start by putting the given equation in standard form:

8x^2 + y^2 + 80x - 6y + 193 = 0

We can begin by moving the constant term to the right side:

8x^2 + y^2 + 80x - 6y = -193

Next, we group the x and y terms together:

(8x^2 + 80x) + (y^2 - 6y) = -193

Now, complete the square for both the x and y terms:

To complete the square for the x terms, we take half of the coefficient of x (80/2 = 40), square it (40^2 = 1600), and add/subtract it within the parentheses:

(8x^2 + 80x + 1600) + (y^2 - 6y) = -193 + 1600

To complete the square for the y terms, we take half of the coefficient of y (-6/2 = -3), square it (-3^2 = 9), and add/subtract it within the parentheses:

(8x^2 + 80x + 1600) + (y^2 - 6y + 9) = -193 + 1600 + 9

Now, simplify both sides of the equation:

(8x + 40)^2 + (y - 3)^2 = 1416

To bring the equation to the standard form, divide both sides of the equation by the constant term on the right side:

((8x + 40)^2)/1416 + ((y - 3)^2)/1416 = 1

Now we have the equation in standard form! Comparing this equation to the standard form equation, we can determine the center, major axis, minor axis, and distance from the center to the foci.

Center:
The center of the ellipse is given by (-h, -k). In this case, h = -40 and k = 3. Therefore, the center is (-(-40), -(3)), which simplifies to (40, -3).

Major Axis:
Since 'a' represents half the length of the major axis, we can find it by taking the square root of 1416 (the denominator of the x term) and dividing it by 2. Thus, a = √(1416)/2.

Minor Axis:
Similarly, 'b' represents half the length of the minor axis, so we take the square root of 1416 (the denominator of the y term) and divide it by 2. Hence, b = √(1416)/2.

Distance from Center to Foci:
The distance from the center to the foci is denoted by 'c'. We can find 'c' using the formula c = √(a^2 - b^2).

Plug in the values of 'a' and 'b' to find 'c':
c = √((√(1416)/2)^2 - (√(1416)/2)^2)

Simplified, this becomes:
c = √(1416/4 - 1416/4)
c = √0
c = 0

Since c = 0, the distance from the center to the foci is zero.

In summary:
- The center of the ellipse is at (40, -3).
- The major axis has a length of √(1416)/2.
- The minor axis has a length of √(1416)/2.
- The distance from the center to the foci is 0.