Find the two equations of the tangent to the circle x^2 + y^2 - 4x - 2y - 20=0 which are parallel to the line 4x+3y=10

after about 30 mins of typing my long detailed explanation, my internet went out and timed out my answer and I was not able to copy my answer because of the time out notification so here is a shortened version of the answer (still correct but with much less explanation)

Using calculus

a line paralell to ax+by=c will be ax+by=d where a=a, b=b and d is a constant

so the paralell lines will have forms 4x+3y=d and 4x+3y=f

find implicit derivitive of circle function

if we solve for where slope is equal to -4/3 (so that it is paralell to other line) we get

3x-4y=2, this is the equation of the line that passes through the circle at the 2 points where the 2 lines we want are

solving for x, we get x=(4y+2)/3

if we subsitute that for x in the circle equation, we get

y^2-2y-8=0

ergo y=-2 and y=4

subsitute that for y to find x in the x=(4y+2)/3 equation

when y=-2, x=-2

when y=4, x=6

now subsitute in to equations to find constants

4x+3y=d, find d

y=-2 and x=-2

4(-2)+3(-2)=d

-8-6=d

-14=d

4x+3y=-14 is one equation

4x+3y=f, find f

y=4, x=6

4(6)+3(4)=f

24+12=f

36=f

4x+3y=36 is another equation

the equations are

4x+3y=-14 and 4x+3y=36