Hello!

First, make this equation a canonical form, (x-h)^2/a^2 + (y-k)^2/b^2 = 1.Then the center is (h, k), the major axis is 2*max(a,b), the minor is 2*min(a,b) and the distance from the center to foci is sqrt(max(a,b)^2 - min(a,b)^2).

To make a canonical form, complete the squares:

4x^2 - 48x = 4*(x^2 - 12x) = 4*((x-6)^2 -36) = 4*(x-6)^2 - 4*36.y^2 - 4y = (y-2)^2 - 4.

The equation becames

4*(x-6)^2 - 4*36 + (y-2)^2 - 4 + 48 = 0 or4*(x-6)^2 + (y-2)^2 = 4*36 + 4 - 48 = 4*(36 + 1 - 12) = 4*25 = 100.

Now divide this by 4*25 to make 1 at the right:

(x-6)^2/25 + (y-2)^2/100 = 1, or(x-6)^2/(5^2) + (y-2)^2/(10^2) = 1.

From here we obtain the answers directly:

a. (6,2)b. 20c. 10d. sqrt(75) = 5*sqrt(3)

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