# 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.