# Equations of Tangent and Normal to the Parabola

Here we list the equations of tangent and normal for different forms of a parabola.

• The euation of tangent to the parabola ${y^2} = 4ax$ at $\left( {{x_1},{y_1}} \right)$ is $y{y_1} = 2a\left( {x + {x_1}} \right)$
• Th eequation of normal to the parabola ${y^2} = 4ax$ at $\left( {{x_1},{y_1}} \right)$ is $y – {y_1} = – \frac{{{y_1}}}{{2a}}\left( {x + {x_1}} \right)$
• If $– \frac{{{y_1}}}{{2a}} = m$ then the equation of a normal line is $y = mx – 2am – a{m^3}$
• The line $y = mx + c$ touches the parabola ${y^2} = 4ax if c = \frac{a}{m}$
• The equation of tangent to the parabola ${y^2} = 4ax$ at $\left( {a{t^2},2at} \right)$ is $x – ty + a{t^2} = 0$
• The equation of normal to the parabola ${y^2} = 4ax$ at $\left( {a{t^2},2at} \right)$ is $y + tx = 2at + a{t^3}$