Equations of Tangent and Normal to the Circle

Here we list the equations of tangent and normal for different forms of a circle and also list the condition of tangency for the line to a circle.

  • The 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}

  • The 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

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

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

  • The 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

  • The 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