Equations of Tangent and Normal to a Parabola

Here we list the equation of tangent and normal for different forms of parabola:

  • Equation 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)

  • Equation 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 equation of 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}
  • Equation of tangent to the parabola {y^2} = 4ax at \left( {a{t^2},2at} \right) is

    x - ty + a{t^2} = 0

  • Equation of normal to the parabola {y^2} = 4ax at \left( {a{t^2},2at} \right) is

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

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