section_path: Analytic results/Tangent lines
section_name: Tangent lines
paragraph_text: Tangent lines
  
  The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x, y) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x, y), so it has the form (x − a) x + (y – b) y = c. Evaluating at (x, y) determines the value of c, and the result is that the equation of the tangent is (x 1 − a) x + (y 1 − b) y = (x 1 − a) x 1 + (y 1 − b) y 1, {\displaystyle (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},} or (x 1 − a) (x − a) + (y 1 − b) (y − b) = r 2. {\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}
  
  If y ≠ b, then the slope of this line is d y d x = − x 1 − a y 1 − b. {\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.}
  
  This can also be found using implicit differentiation.
  
  When the centre of the circle is at the origin, then the equation of the tangent line becomes x 1 x + y 1 y = r 2, {\displaystyle x_{1}x+y_{1}y=r^{2},} and its slope is d y d x = − x 1 y 1. {\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}