# Equations of Tangent and Normal to a Ellipse

Here we list the equation of tangent and normal for different forms of ellipse also we define parallel chords and condition of tangency of an ellipse:

• Equation of tangent to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at $\left( {{x_1},{y_1}} \right)$ is

• Equation of normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at $\left( {{x_1},{y_1}} \right)$ is

• Equation of tangent to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at $\left( {a\cos \theta, b\sin \theta } \right)$ is

• Equation of normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at $\left( {a\cos \theta, b\sin \theta } \right)$ is

• The locus of middle points of parallel chords of an ellipse is the diameter of ellipse and has equation

• The condition for $y = mx + c$ to be the tangent to the ellipse is