# Distance problems?

• Dec 9th 2012, 07:31 PM
camjenson
Distance problems?
y=x^(1/2). what's the shortest distance from that curve to the point (1/2,0)?
• Dec 9th 2012, 08:08 PM
MarkFL
Re: Distance problems?
Let $\displaystyle \left(x,x^{\frac{1}{2}} \right)$ be an arbitrary point on the curve. It will be simpler to minimize the square of the distance between this arbitrary point and the given fixed point. Can you state the square of this distance?
• Dec 9th 2012, 08:15 PM
camjenson
Re: Distance problems?
would this use the distance formula? or am i way off?
• Dec 9th 2012, 08:17 PM
MarkFL
Re: Distance problems?
You are spot on! Since our objective function is a distance, then that's a great place to start!
• Dec 9th 2012, 08:45 PM
camjenson
Re: Distance problems?
so how do i use that? what do i plug in?
• Dec 9th 2012, 09:15 PM
MarkFL
Re: Distance problems?
We have two points:

$\displaystyle \left(x,x^{\frac{1}{2}} \right)$ and $\displaystyle \left(\frac{1}{2},0 \right)$.

Now, the distance formula tells us the distance $\displaystyle d$ between the two points $\displaystyle P_1(x_1,y_1)$ and $\displaystyle P_2(x_2,y_2)$ is:

$\displaystyle d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

So, choose either of our two points to be $\displaystyle P_1$ and the other as $\displaystyle P_2$ and plug into the equivalent:

$\displaystyle d^2=(x_2-x_1)^2+(y_2-y_1)^2$

Then simplify and optimize by differentiation. Be sure to demonstrate you have minimized by using an appropriate test.