<s>[INST] How to find points of tangency I need to find the points of tangency on a circle (x^2+y^2=100) and a line y=5x+b the only thing I know about b is that it is negative. This line runs parallel to the line y=5x+7. I need to find the points of tangency between the line y=5x+b and the circle. 
 [/INST] The tangent will be perpendicular to the radius  $y = -\frac 15 x.$
This line intersects the circle.
$x^2 + (-\frac 15 x)^2 = 100\\
\frac {26}{25} x^2  = 100\\
x = \pm 50\frac {\sqrt {26}}{26},y = \mp 10 \frac{\sqrt {26}}{26}$
There are two points of tangency.
$(y+10\frac {\sqrt{26}}{26}) = 5(x-50\frac {\sqrt{26}}{26})\\
(y-10\frac {\sqrt{26}}{26}) = 5(x+50\frac {\sqrt{26}}{26})$
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