Equations of Tangent and Normal to a Circle

Here we list the equation of tangent and normal for different forms of circle and also list the condition of tangency for the line to the circle:

  • Equation of tangent to the circle {x^2} + {y^2} =  {a^2} at \left( {{x_1},{y_1}}  \right) is

    x{x_1} + y{y_1} = {a^2}

  • Equation of normal to the circle {x^2} + {y^2} = {a^2} at \left( {{x_1},{y_1}} \right) is

    y{x_1} - x{y_1} = 0

  • Equation of tangent to the circle {x^2} + {y^2} + 2gx  + 2fy + c = 0 at \left(  {{x_1},{y_1}} \right) is

    x{x_1} +  y{y_1} + g\left( {x + {x_1}} \right) + f\left( {y + {y_1}} \right) + c = 0

  • The condition of tangency for a line y = mx + c to the circle {x^2} + {y^2} = {a^2} is

    c =  \pm a\sqrt {1 + {m^2}}

    and equation of tangent to the circle {x^2} + {y^2} = {a^2} is

    y = mx \pm a\sqrt {1 + {m^2}}

  • Equation of tangent to the circle {x^2} + {y^2} =  {a^2} at \left( {a\cos \theta ,a\sin  \theta } \right) is

    x\cos  \theta + y\sin \theta = a

  • Equation of normal to the circle {x^2} + {y^2} = {a^2} at \left( {a\cos \theta ,a\sin \theta }  \right) is

    x\sin \theta - y\cos \theta = 0