Equation of Tangent to a Circle

We can find the equation of a tangent to a circle in the following way.
Suppose we have a point  
  on a circle and a tangent at that point. The centre of the circle is at the point  
  say, so the gradient of the this particular radius is  
  and the tangent, being at right angles to the radius at the point where they meet on the circle, has gradient  
\[m_{TANGENT}=- \frac{1}{m_{RADIUS}}\]
The equation of the tangent is then  
\[y-y_1 =m_{TANGENT}(x-x_1)\]

Example: A circle has centre  
. Find the equation of the tangent to the circle at the point  
The gradient of the radius drawn from  
\[m_{RADIUS}=\frac{6-2}{6-3}= \frac{4}{3}\]
The gradient of the tangent is then  
\[m_{TANGENT} = -\frac{1}{4/3}=- \frac{3}{4}\]
The equation of the tangent is  
\[y-2=- \frac{3}{4} (x-3)=- \frac{3}{4} x+\frac{9}{4}\]
\[y=-\frac{3}{4}x +\frac{9}{4}+2=-\frac{3}{4}x +\frac{17}{4}\]

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