# Thread: Finding equation of a line

1. ## Finding equation of a line

i don't understand this AT ALL. oh calculus.
would it be 10x-5, while 10 is the slope...gah i have no clue X_X plz help

Find an equation of the line that is both normal to the given parabola and also parallel to the given line

y = 5 x 2 - 5 x + 6
x -
5 y = 6

2. $\displaystyle y = 5x^2 - 5x + 6$

$\displaystyle y' = 10x - 5$

the gradient function is 10x - 5.

the line
$\displaystyle x - 5y = 6$
$\displaystyle x - 6 = 5y$
$\displaystyle \frac{x}{5}- \frac{6}{5} = y$

the gradient of the wanted line must be 6/5, and hence 6/5 is the normal to the parabola. the gradient of the tangent to the parabola must therefore be -5/6

$\displaystyle \frac{-5}{6}= 10x - 5$

Solve for x

$\displaystyle 4\tfrac{1}{6} = 10x$

$\displaystyle x= \frac{5}{12}$

when $\displaystyle x= \frac{5}{12}$ $\displaystyle y = 4\tfrac{113}{144}$

Equation of the line

$\displaystyle \frac{y - 4\tfrac{113}{144}}{x - \frac{5}{12}} = \frac{6}{5}$

cross multiply and you get your answer (i think)