# Thread: Point on Circle Closest to Another Point on Circle

1. ## Point on Circle Closest to Another Point on Circle

Hi all ---

I understand the circle equations. And I understand the vector joining the two centers of the circles is just ---
$[1,1] - [5, 4] = [-4, -3]$ so its norm is $\sqrt{(-4)^2 + (-3)^2}$

But I don't understand the $\frac{2}{5}(-4, -3)$ in the last line at all. Where does it come from? What does it mean?

Thanks a lot ---

2. ## Re: Point on Circle Closest to Another Point on Circle

distance between the two centers is 5

radius of the first circle is 2

point on the first circle closest to the second lies 2/5 of the total distance from center to center.

start position (center of the first circle) + 2/5 of the direction vector from center to center

(5,4) + (2/5)<-4,-3>

3. ## Re: Point on Circle Closest to Another Point on Circle

Originally Posted by skeeter
distance between the two centers is 5

radius of the first circle is 2

point on the first circle closest to the second lies 2/5 of the total distance from center to center.
Hi skeeter ---

Thanks for answering. The part above I get. But the rest of your post I don't, sadly.

I drew a sketch ---

Because $[5,4]$ = vector to the center of the large circle is NOT on the same line as $[4, 3]$ = distance between the two centers ---

how does [ $5, 4] - (2/5)[4, 3]$ give us the point we want?

I get your post if the green and red lines are collinear, but that's not the case here?

Thanks a lot ---

start position (center of the first circle) + 2/5 of the direction vector from center to center

(5,4) + (2/5)<-4,-3>

4. ## Re: Point on Circle Closest to Another Point on Circle

think of it as adding two position vectors ...

$(5i + 4j) + \frac{2}{5}(-4i - 3j)$

the sum of the two vectors is the position on the large circle closest to the center of the small circle.

5. ## Re: Point on Circle Closest to Another Point on Circle

Hi skeeter ---

I get how to do arithmetic with vectors, but I still somehow don't understand why $[5,4] - (2/5)[4, 3]$ works, probably because $[4, 3]$ and $[5,4]$ are NOT collinear.

Here's what I'm thinking --- $(2/5)[4, 3]$ gives the radius of the big circle right on $[4, 3]$ - the vector joining the two circles's centers.

But doing $[5,4] - (2/5)[4, 3]$ looks to me like we're getting the part/rest of the red vector that's outside the big circle. But the red vector isn't on $[4, 3]$ - so it wouldn't give the point on the big circle closest to the small circle?

And I know that the part of the green and red vectors inside the big circle have the same length in the big circle, because they are both radii.

6. ## Re: Point on Circle Closest to Another Point on Circle

Originally Posted by mathminor827
I get how to do arithmetic with vectors, but I still somehow don't understand why $[5,4] - (2/5)[4, 3]$ works, probably because $[4, 3]$ and $[5,4]$ are NOT collinear.
The set of points $(5-t,4-3t)~0\le t\le 1)$ is the line segment between the centers.
When $t=0$ we get $(5,4).$
When $t=1$ we get $(4,1).$
When $t=\frac{2}{5}$ we get the point we want.

7. ## Re: Point on Circle Closest to Another Point on Circle

Originally Posted by Plato
The set of points $(5-t,4-3t)~0\le t\le 1)$ is the line segment between the centers.
When $t=0$ we get $(5,4).$
When $t=1$ we get $(4,1).$
When $t=\frac{2}{5}$ we get the point we want.
Hi Plato ---

Thanks for answering. But aren't you working backwards from the answer? I mean - I understand what you're saying.

But I'm stuck on the idea or process to actually get the answer - I'm not asking about the actual values of the coordinate.

Thanks ---

8. ## Re: Point on Circle Closest to Another Point on Circle

Originally Posted by mathminor827
But aren't you working backwards from the answer? I mean - I understand what you're saying.
Absolutely not!
The length of that line segment between centers is 5.
The radius of the circle centered at (5,4) is 2.
From $(5,4)$ we want a point that is $\frac{2}{5}$ the way between the centers.