<s>[INST] A line through the point P(8, -7) is a tangent to the circle C at the point T. Find T 
Circle C equation $(x+5)^2+(y-9)^2=25$
  A line through the point P(8, -7) is a tangent to the circle C at the point T.
  Find T.

I tried simultaneous equations:
1. $(x+5)^2+(y-9)^2=25$
2. $y = m(x-8)-7$
3. differentiate circle: $\dfrac {dy} {dx} = \dfrac{-(x+5)} {y-9}$
where x, y  represent co-ordinates of T  
But it does not seem to be getting anywhere.
By using first 3 equations I get horrible mix of $x, x^2, x^3, y^2, y, xy$ 
Many thanks in advance
 [/INST] plug in $ m= \dfrac {dy} {dx} = \dfrac{-(x+5)} {y-9}$  you get
$$y = \dfrac{-(x+5)(x-8)} {y-9}-7 \Rightarrow (y+7)(y-9) = -(x+5)(x-8) 
  $$
so your answer will be the solution of the following system ..
$\begin{array}{r c l} (y+7)(y-9) + (x+5)(x-8) &=&0 \\ (x+5)^2+(y-9)^2 &=&25 \end{array}$
you just have to expand both equations and subtract them to cancel all the quadratic terms and you will be left with the equation of a line.
The solutions to your problem will be the points of intersection between that line and the circle 
$(x+5)^2+(y-9)^2 =25 $
</s>