# Math Help - Finding point where normal line interesects parabola

1. ## Finding point where normal line interesects parabola

The problem is: at what point does the normal to at intersect the parabola a second time?

$y ' = 8x +3$

Then can I just take the slope of 8, inverse it to be $-1/8$ and then put it in point slope form, $(y-9)=-1/8(x-1)$
This ends up being $y = -1/8x + 73/8$.

Then do I make it $-1/8x + 73/8 = 4x^2 + 3x + 2$ ? The math gets pretty hairy after that, and I just would like to know if I'm going in the right direction!

2. ## Re: Finding point where normal line interesects parabola

Originally Posted by AZach
The problem is: at what point does the normal to at intersect the parabola a second time?

$y ' = 8x +3$

Then can I just take the slope of 8, inverse it to be $-1/8$ and then put it in point slope form, $(y-9)=-1/8(x-1)$
This ends up being $y = -1/8x + 73/8$.

Then do I make it $-1/8x + 73/8 = 4x^2 + 3x + 2$ ? The math gets pretty hairy after that, and I just would like to know if I'm going in the right direction!
No you CAN'T just take the slope as being 8. The derivative gives you a FUNCTION which gives you the gradient of your curve at ANY point. It CHANGES as you change your point. The gradient at x = 1 is 8(1) + 3 = 11.

3. ## Re: Finding point where normal line interesects parabola

Ah, that definitely explains my first problem.

Then the tangent line is $y = 11x + 9$ and the normal line is $(y-9) = -1/11(x-1)$ or $y = -1/11x + 100/11$

Next step, I substitute $-1/11x + 100/11 = 4x^2 + 3x + 2$ since one of the equations is already equal to y.

Multiply across by 11 to clear fractions $-x + 100 = 44x^2 + 33x + 22$ -> $0 = 44x^2 + 34x - 78$ -> Divide the equation by 2 $0 = 22x^2 + 17x -39$.

My next question, is the quadratic formula the next way to go? I've already got it computed, but I'm not sure what the answer means or if it's correct. I know the objective is to find the perpendicular line to the parabola and find the second intersection point. So, the quadratic formula should give me the x coordinates for the intersection points on the parabola, how do I get the f(x) coordinates?

4. ## Re: Finding point where normal line interesects parabola

Okay, so I get it now. I use the quad. formula and got x = 1 (makes sense, this is the point given in the question and corresponds with a output of 9), and x = -1.772727273. I plugged the latter into the original function of the parabola and got 9.252066119, so the normal line intersects the parabola a second time at (-1.772727273, 9.252066119). (1,9) is where it intersects the first time.

Is this kind of information related to finding tension in mechanics? Like if there's a swing or pendulum, could finding the intersection points of the perpendicular tell where the machine is stressed or receiving any kind of instantaneous change?