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 \[y + tx = 2at + a{t^3}\]