The circle C has centre (3, 1) and passes through the point P(8, 3) - Edexcel - A-Level Maths Pure - Question 6 - 2008 - Paper 2

To find the equation of the tangent at point P(8, 3), we first need the gradient of the radius:

1. Gradient of radius:

   The radius connects (3, 1) and (8, 3) and the gradient is given by:

   $m_{radius} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{8 - 3} = \frac{2}{5}$

2. Gradient of tangent:

   The tangent at P will have a negative reciprocal gradient:

   $m_{tangent} = -\frac{1}{m_{radius}} = -\frac{5}{2}$

3. Use point-slope form of the line:

   Using the point P(8, 3):

   $y - 3 = -\frac{5}{2}(x - 8)$

4. Rearranging to the required form:

   Expanding this gives:

   $y - 3 = -\frac{5}{2}x + 20 \implies 5x + 2y - 26 = 0$

   Final form:

   $5x + 2y - 26 = 0$