Thread: optimization - distance between points on a graph

1. optimization - distance between points on a graph

Find the points on the graph of $y=9-x^2$ that is closest to the point $(0, 3)$.

2. In general, the distance between two points is

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

Use:

$(x_1,y_1)=(x,9-x^2)$ <-- this is a point on the curve
$(x_2,y_2)=(0,3)$

Then we have the distance between a point on the curve and the point $(0,3)$:

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

$=\sqrt{(x-0)^2+(9-x^2-3)^2}$

$=\sqrt{x^2+(6-x^2)^2}$

$=\sqrt{x^2+36-12x^2+x^4}$

$=\sqrt{x^4-11x^2+36}$

Our goal is to find the minimum value of this, so we find the derivative and set it equal to 0 and solve for x.

Can you take it from here?

3. Yes, thank you!

One question though, in the general formula for the distance between two lines, does it matter which function i assign to be $(x_1,y_1)$ and $(x_2,y_2)$?

4. Originally Posted by shawli
Yes, thank you!

One question though, in the general formula for the distance between two lines, does it matter which function i assign to be $(x_1,y_1)$ and $(x_2,y_2)$?
No, it works either way. If you need to be convinced, try switching it around and see that you arrive at the same expression.