(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

There is a point (4,2), from which I have to find the least distance to the parabola y^2=8x. I.e, the least distance connecting the point and the parabola.

2. Relevant equations

yy1=2a(x+x1)

Perpendicular distance = mod[(ax1+by1+c)/sqrt(a^2+b^2)]

3. The attempt at a solution

I already know the solution to this one. Our professor said the geometric way of solving this one may not seem easy. So we did this by using "Applications of Derivatives", i.e. Minima.

Somehow, I found a geometrical way of solving this. It goes like this, if I take a point P(x1,y1) on the parabola such that, when a tangent is drawn to this point, a normal from the point of contact to this tangent passes through (4,2). By using the formula to find distance between a point and line, i.e. the point (4,2) and the tangential line yy1=2a(x+x1), and then using y1^2=2ax1, we can get a quadratic equation in x1 or y1, which one solving matches the given solution.

I don't know how I landed on this, but why is the above true? The tangent I've considered is quite special, considering the normal to it from a point coincides with the point of contact. How does this deliver the least distance? Can someone shed some light on related facts?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Least distance from a point to a parabola

**Physics Forums | Science Articles, Homework Help, Discussion**