Car traveling along parabola.. at what point will headlights illuminate point (10,5)?

Question

I'm kinda confused on this problem
A car is traveling at night along a highway shaped like the parabola y=(x^2)/10 from left to right. At what point on the highway will the headlights illuminate the point (10,5)?

Thanks

Answer 1

The beam of the headlights will be tangent to the curve. In other words the slope of the beam will be the derivative of the curve.

So slope of beam = m = d(x^2/10)/dx = x/5

The beam will then be a straight line having this slope and going through the point (10,5). So:
y = mx + b giving 5 = 10m + b

Call X and Y the point on the parabola , so:
Y = X^2/10
This point must also be on the straight line:
Y = (X/5)X + b = X^2/5 + b and also 5 = 2X + b
The first is using the point on the parabola and the second is from the point the beam goes through.

X^2/10 = X^2/5 + b
b = -X^2/10
5 = 2X - X^2/10 or X^2 - 20X + 50 = 0
X = 2.929 and Y = 0.857

Answer 2

Let's get rid of the car and put this in mathematical terms. You are looking for the point on the parabola at which a line through the point (10,5) would be tangent.

Slope of a tangent line at any point is the first derivative of the function, so do that.

Then use the two-point method from (X,X^2/10) to (10,5), set that equal to the number you got above and solve.