# Point on a parabola where the tangent intersects a point

1. Oct 19, 2009

### sp09ta

1. A car is traveling on a highway shaped like a parabola with its vertex at the origin. The car begins at a point 100m west and 100m north of the origin and is traveling easterly. There is a statue 100m east and 50m north of the origin. At what point on the highway will the cars headlights illuminate the statue.

2. d/dx x^n = x*x^(n-1)
(y2-y1)/(x2-x1)=m

3. The car begins at (-100,100) and travels on the domain [-100,inf).

let f(x) represent the path of the car, thus f(x) will represent the cars headlights.

f(x)=(1/100)*x^2 and f'(x)=(1/50)*x

We need to find point P, who's tangent line intersects (100,50).

Since the equation of the tangent is y=mx+b, and m=f'(x), then (50-y)/(100-x)=(1/50)x

I then solved for y to recieve y=(x^2+100x+2500)/50

and for y=0, x=50.

So sub y=50 into my original eqn:

Thus, 50=(1/100)*x^2 so, 50sqrt(2)=x

And the statue will be illuminated by the cars headlights when the car is at (50*sqrt(2), 50)?

Does this look correct?

Last edited: Oct 19, 2009