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 \[x\sin \theta – y\cos \theta = 0\]