Given Equation: 4x^2 + y^2 - 8x - 2y + 1 = 0

To find out the length of major and minor axes, latus rectum, eccentricity, coordinates of the centre, foci, and equation of the directrix, we need to convert the given equation into the standard form of an ellipse.

Step 1: Grouping the terms
4x^2 - 8x + y^2 - 2y + 1 = 0

Step 2: Completing the square for x-terms
4(x^2 - 2x) + y^2 - 2y + 1 = 0
4(x^2 - 2x + 1) + y^2 - 2y + 1 = 4
4(x - 1)^2 + y^2 - 2y + 1 = 4

Step 3: Completing the square for y-terms
4(x - 1)^2 + (y^2 - 2y) + 1 = 4
4(x - 1)^2 + (y^2 - 2y + 1) + 1 - 1 = 4
4(x - 1)^2 + (y - 1)^2 = 4

Step 4: Simplifying the equation
(x - 1)^2/1 + (y - 1)^2/4 = 1

The given equation is now in the standard form of an ellipse: (x - h)^2/a^2 + (y - k)^2/b^2 = 1

Where the center of the ellipse is at (h, k) = (1, 1).

Step 5: Finding the length of major and minor axes
The length of the major axis is 2a, and the length of the minor axis is 2b.

From the equation, we can see that a^2 = 1 and b^2 = 4.

Therefore, the length of the major axis is 2a = 2(1) = 2 units, and the length of the minor axis is 2b = 2(2) = 4 units.

Step 6: Finding the latus rectum
The latus rectum is given by the formula 2b^2/a.

Substituting the values, we get latus rectum = 2(4^2)/1 = 32 units.

Step 7: Finding the eccentricity
The eccentricity of the ellipse is given by the formula c/a, where c is the distance between the center and the foci.

From the equation, we can see that a^2 = 1, so a = 1.

To find c, we use the formula c^2 = a^2 - b^2.

Substituting the values, we get c^2 = 1 - 4 = -3.

Since c^2 is negative, the eccentricity is imaginary, and hence the ellipse is degenerate.

Step 8: Finding the coordinates of the foci
Since the eccentricity is imaginary, there are