Question 824807
3x-4y-2=0 same as -4y=-3x+2, y=(3/4)x-1/2.
General point is (x,(3/4)x-1/2).


3x-4y+k=0 same as -4y=-3x-k, y=(3/4)x+k/4.
General point is (x, (3/4)x+k/4).


You want each of these line to be the same distance from point (-1,-5).  Using Distance Formula, these distances each are:


{{{sqrt((x+1)^2+((3/4)x-1/2+5)^2)}}}   and    {{{sqrt((x+1)^2+((3/4)x+k/4+5)^2)}}}
{{{sqrt((x+1)^2+((3/4)x+11/2)^2)}}}     and     {{{sqrt((x+1)^2+((3/4)x+(k+20)/4)^2)}}}
{{{sqrt((x+1)^2+((3x+22)/4)^2)}}}     and     {{{sqrt((x+1)^2+((3x+k+20)/4)^2)}}}


We must look very carefully at each expression and understand that {{{(3x+22)/4}}} must be equal to {{{(3x+k+20)/4}}}.  They MUST correspond as equivalent!


{{{(3x+22)=(3x+k+20)}}}
{{{highlight(highlight(k=2))}}}