Where Does the Normal Line Intersect the Parabola Again?

[volc]

Here is what I found, look at the end to see what my question is:

At what point does the normal line to the graph of y= -4 - 3x + 4x^2 at (1, -3 ) intersect the parabola a second time?

So far I have tried to find the derivative of the parabola which is:

8x-3 (this corresponds to the slope of the tangent line). Therefore now I replaced x by 1 because of the point I am given. This gives:

8(1)-3 = 5

So that is the slope of the tangent line. Now to know the equation of this line I do:

y=5x+b , replace x and y by the point I am given and solve for b.

-3=5(1)+b
b=-8

Thus, the equation is: y=5x-8

The normal is perpendicular to the tangent at (1,-3) so I take the negative reciprocal of the slope to find the slope of the normal:

5 becomes -1/5 (reciprocal). And now I get y=-1/5+b for the normal. Replace x and y by the point (1,-3):

y=-1/5x+b
-3=-1/5x+b
-3=-1/5(1)+b
-14/5=b

The equation for the normal is: y=-1/5x-14/5

Now I want to find when this line and the parabola meet so:

-1/5x-14/5 = -4 - 3x + 4x^2
0 = -6/5 -14/5x + 4x^2
Using (-b+/-sqrt(b^2-4ac))/2a
x1 = 1
x2 = -3/10

x1 = 1 I already know because that is the point where the normal, the tangent, and the parabola intersect. And x2 = -3/10 is the other point where only the parabola and the normal intersect.

To find the y-coordinate of -3/10 I just insert it into the equation and get:

y=-1/5(-3/10)-14/5
y= -137/50

Thus the other point is: (-3/10,-137/50)

I would just like to know if the above calculations are correct and that I am doing the proper steps.

Thanks a lot,

Cedrick O'Shaughnessy

 

[StatusX]

That all looks good. I'm just not sure why you found the equation for the tangent line, since it doesn't seem like you needed it.

 

[quicknote]

Your calculations and steps seem correct. You have correctly found the slope of the tangent line and used it to find the equation of the tangent line. You also found the slope of the normal line and used it to find the equation of the normal line. Finally, you solved for the points of intersection between the normal line and the parabola. Your answer of (-3/10, -137/50) as the second point of intersection is also correct.

One thing to note is that in your equation of the normal line, you have a negative sign in front of the slope (-1/5x - 14/5). This is because the slope of the normal line is perpendicular to the slope of the tangent line, so it should be the negative reciprocal. However, this does not affect your calculations or final answer.

Overall, your approach and calculations are correct. Keep up the good work!