Circle

Rewriting the equation of the circle as:
x2 + y2 - 4x - 8y + 16 = 0
The tangent to the circle at point (h, k) lying on the circle is:
hx + ky - 2(h + x) - 4(k + y) + 16 = 0
With it passing through the point (-4, 1), we have:
-4h + k - 2(h - 4) - 4(k + 1) + 16 = 0
k = -2h + 20/3
Sub it back to the circle's equation:
h2 + (-2h + 20/3)2 - 4h - 8(-2h + 20/3) + 16 = 0
h2 + 4h2 - 80h/3 + 400/9 - 4h + 16h - 160/3 + 16 = 0
5h2 - 44h/3 + 64/9 = 0
h = (22 + 2√41)/15 or (22 - 2√41)/15
k = (56 - 4√41)/15 or (56 + 4√41)/15
So the points of contact are:
[(22 + 2√41)/15, (56 - 4√41)/15] and [(22 - 2√41)/15, (56 + 4√41)/15]

Besides the above method, I now give you another method in case
you don't remember the formula for the tangent of a circle.

Let P(a, b) be a point such that the line passing through the centre of
circle and P is perpendicular to the line passing through (- 4, 1) and P.

Then, by considering their slope, we have :
[(b - 4)/(a - 2)] * [(b - 1)/(a + 4)] = -1
_______________b^2 - 5b + 4 = - (a^2 + 2a - 8)
____________(b - 5/2)^2 - 9/4 = - (a + 1)^2 + 9
_______(a + 1)^2 + (b - 5/2)^2 = 45/4

Now, let Q(h, k) be the required point.
Then, we have to solve the simultaneous equations :
(h - 2)^2 + (k - 4)^2 = 4 ----------------- (1)
(h + 1)^2 + (k - 5/2)^2 = 45/4 ---------- (2)

From (1) : h = sqrt[4 - (k - 4)^2] + 2 or - sqrt[4 - (k - 4)^2] + 2 ----- (3)
Sub. h = sqrt[4 - (k - 4)^2] + 2 into (2) :
______{sqrt[4 - (k - 4)]^2 + 2 + 1}^2 + (k - 5/2)^2 = 45/4
4 - (k - 4)^2 + 6 sqrt[4 - (k - 4)^2] + 9 + (k - 5/2)^2 = 45/4
________________________6 sqrt[4 - (k - 4)^2] = 8 - 3k
______________________________4- (k - 4)^2 = (4/3 - k/2)^2
_____________________4 - k^2 + 8k - 16 = 16/9 - (4/3)k + (k^2)/4
_______________________45 k^2 - 336k + 496 = 0
__________________________________k = [56 +/- 4 sqrt(41)]/15 ----- (4)

Sub. (4) into (1) :
We can get h = [22 - 2 sqrt(41)]/15 when k = [56 + 4 sqrt(41)]/15
and h = [22 + 2 sqrt(41)]/15 when k = [56 - 4 sqrt(41)]/15

So, the points of contact of the circle with the tangents are :
{[22 - 2 sqrt(41)]/15, [56 + 4 sqrt(41)]/15} and
{[22 + 2 sqrt(41)]/15, [56 - 4 sqrt(41)]/15}