4x² + y² - 8x - 2y + 1 = 0

Get the constant off the left side:

    4x² + y² - 8x - 2y = -1

Swap the 2nd and 3rd terms

    4x² - 8x + y² - 2y = -1

Out of the first two terms factor out the coefficient of x²
which is 4

  4(x² - 2x) + y² - 2y = -1

We complete the square inside the parentheses:
Multiply the coefficient of x, which is -2, by ,
getting -1, Then square that result, (-1)² = 1. So we add
1 inside the parentheses, getting 4(x² - 2x + 1), which,
because of the 4*1 in front amount to adding 4 to the left
side so we have to add 4 to the right side:

  4(x² - 2x + 1) + y² - 2y = -1 + 4

Next we complete the square for the y terms:
Multiply the coefficient of y, which is -2, by ,
getting -1, Then square that result, (-1)² = 1. So we add
1 to both sides:

  4(x² - 2x + 1) + y² - 2y + 1 = -1 + 4 + 1

Now we factor "x²-2x+1" as (x-1)² and "y²-2y+1" as "(y-1)²"
and combine the numbers on the right as 4

  4(x - 1)² + (y - 1)² = 4

Then we divide through by 4 to get 1 on the right

  

  

Compare that to the standard form:

  

since a > b, so a² > b² in an ellipse. a=1, b=2

So this is an ellipse with center (h,k) = (1,1),

That's one question answered:  center = (1,1)

Major axis vertical which is 2a or 2(2) or 4

Minor axis horizontal which is 2b or 2(1) = 2

So we draw the center (1,1) and the major and minor axes,
respective 4 and 2 units long with the center (1,1) as the
midoint of each.



Then we draw in the ellipse:



The vertices are at the top and bottom.  They are (1,3) and (1,-1)



We have to calculate the focal points which are (h,k±c) inside 
the ellipse on the major axis:

First we calculate c from

c² = a² - b²

c² = 2² - 1²

c² = 4 - 1

c² = 3
      _
c = ±Ö3
                      _            _
So the foci are (1,1+Ö3) and (1,1-Ö3). They are the two points plotted
below inside the ellipse on the major axis.



Edwin