Question 310361
Let's use implicit differentiation to find the derivative at that point.
{{{(x-9)^2+(y-2)^2=25}}}
Differentiating,
{{{2(x-9)dx+2(y-2)dy=0}}}
{{{(y-2)dy=(9-x)dx}}}
{{{dy/dx=(9-x)/(y-2)}}}
The value of the derivative equals the slope of the tangent line at that point.
So at (12,-2),
{{{dy/dx=(9-12)/(-2-2)}}}
{{{dy/dx=-3/-4=3/4}}}
Use the point-slope form of a line,
{{{y-yp=m(x-xp)}}}
{{{y-(-2)=(3/4)(x-12)}}}
{{{y+2=(3/4)x-9}}}
{{{y=(3/4)x-11}}}
{{{ drawing( 300, 300, -4, 15, -8, 10, grid(1),circle(12,-2,.4),graph( 300, 300, -4, 15, -8, 10, (3/4)x-11,2+sqrt(25-(x-9)^2), 2-sqrt(25-(x-9)^2))) }}}