# If my tangent line at point (4,8) has the equation y=5x/6 - 9, what is the equation of the normal line at the same point?

Oct 11, 2014

The normal line is the line that is perpendicular to the the tangent line.

If the slope of a line is $m$ then the slope of the perpendicular line is $- \frac{1}{m}$, this is also known as the negative reciprocal.

The given equation is $y = \frac{5}{6} x - 9$ the slope is $\frac{5}{6}$ so the slope of the normal is $- \frac{6}{5}$.

The point $\left(x , y\right) \to \left(4 , 8\right)$

$y = m x + b \to$ Substitute in the values of $m$, $x$ and $y$

$8 = - \frac{6}{5} \left(4\right) + b$

$8 = - \frac{24}{5} + b$

$\frac{24}{5} + 8 = b$

$\frac{24}{5} + \frac{40}{5} = b$

$\frac{64}{5} = b$

The equation of the normal line is $\to y = - \frac{6}{5} x + \frac{64}{5}$