# Thread: Equation of 2 lines

1. ## Equation of 2 lines

I need some help, this is from a non-calculator wkst.

1) Find the equations of both lines through (2, -3) that are tangent to the parabola $\displaystyle y=x^2+x$

2. Originally Posted by JBknights10
I need some help, this is from a non-calculator wkst.

1) Find the equations of both lines through (2, -3) that are tangent to the parabola $\displaystyle y=x^2+x$
let $\displaystyle (x,y)$ be a point on the tangent line

then the slope between $\displaystyle (x,y)$ and $\displaystyle (2,-3)$ is given by $\displaystyle m = \frac {y_2 - y_1}{x_2 - x_1} = \frac {y + 3}{x - 2} = \frac {x^2 + x + 3}{x - 2}$

but this formula for the slope must be equal to the formula for the derivative. since the derivative also gives the slope of the tangent line. so that

$\displaystyle y' = 2x + 1 = \frac {x^2 + x + 3}{x - 2}$

solving $\displaystyle 2x + 1 = \frac {x^2 + x + 3}{x - 2}$, we find the solutions to be $\displaystyle x = 5$ or $\displaystyle x = -1$

so, the slopes of our two tangent lines are given by $\displaystyle y'(5) = 11$ and $\displaystyle y'(-1) = -1$

thus, our two lines are:

(1) the line passing through (2,-3) with slope 11

and

(2) the line passing through (2,-3) with slope -1

i leave it to you to find these lines