Find Equation of Tangent Line to Parabola

Example: Find an equation of tangent to the parabola {y^2}  = 13x parallel to the line 7x - 9y +  11 = 0.
Solution: The given equation of 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 parabola with the general equation of parabola {y^2} = 4ax.
After comparing with standard equation parabola we have a = \frac{{13}}{4}
Now to find slope of given line is m =  -  \frac{7}{{ - 9}} = \frac{7}{9}
The required equation of 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 tangent to the parabola.

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