Find the Equation of the Tangent Line to Parabola

Example: Find the equation of the tangent to the parabola {y^2} = 13x parallel to the line 7x - 9y + 11 = 0.

Solution: The given equation of a parabola can be written in the form:

{y^2} = 4\left( {\frac{{13}}{4}} \right)x\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

Compare this equation of a parabola with the general equation of a parabola {y^2} = 4ax.

After comparing with the standard equation of a parabola we have a = \frac{{13}}{4}

Now to find the slope of a given line we use m = - \frac{7}{{ - 9}} = \frac{7}{9}

The required equation of the tangent to the parabola is given as

\begin{gathered} y = mx + \frac{a}{m} \\ \Rightarrow y = \frac{7}{9}x + \frac{{\frac{{13}}{4}}}{{\frac{7}{9}}} = \frac{7}{9}x + \frac{{117}}{{28}} = \frac{{196x + 1053}}{{252}} \\ \Rightarrow 196x - 252y + 1053 = 0 \\ \end{gathered}

This is the equation of the tangent to the parabola.