# 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 $x\sin \theta – y\cos \theta = 0$