Question

Straight forward calculus question- but I'm stuck!

Find the points on the circle x^2+y^2=9 that are closest to and farthest from (13,17)

Thanks :)

Answer 1

\[x^{2}+y^{2}=9\]

Answer 2

(13,17) is an external point. My approach would be to find equation for distance between this point and (x,y) (a point on the circle) and use \[D ^{2}\] to make it easier- then find derivative of this and set =0 to find min and max.

But where I come unstuck is how to ensure this point lies on the circle...

Answer 3

This is very similar to finding the closes point of a line to another one. In which case I inverse the slope of the line use the point given to find a function of that line set the y equal and then solve for x.....but finding the tangent line requires to know the point already closest to the point given to you so that's what Im confused about. Sorry if this confuses you but I'm just throwing out information and hopefully something sparks

Answer 4

Thanks Romero.

Answer 5

y=17/13.x (line)
intercept it with circle

Answer 6

Thank you RaphaelFilgueiras! Are you saying the closest and furthest points will exist on a line through the origin...? What theorem does this come from? As in how do I justify that in an answer? :)

Answer 7

It has to. The closest and farthest are a reflection of each other.

Answer 8

just for the near,the minimum distance of a pint to a line is a perpenducular connecting them

Answer 9

Ok great, and the max distance, as romero pointed out, is just reflected? Is that a Euclidean theorem that you mentioned?

Answer 10

yes

Answer 11

the max i dont know yet

Answer 12

I imagine the furthest you could get from a point on the circle through a chord is to travel the diameter...so the max must also lie on the line y=17/13x you gave. Thoughts?

Answer 13

Wait you have this equaiont: y=17/13.x (line)
and the one for the circle.

Just set them up together ( solve for y on the circle)
join the equations and solve for x.

You will have two solutions. The closest point and the farthest point and that's because you have x^2

Answer 14

Does this sound right or am I wrong?

Answer 15

Yes, sounds logical. I got closest point (1.8224, 2.383) and furthest point (-1.8224, -2.383). I just wish I could double check somehow.

Answer 16

Draw it? lol

Answer 17

Drawing it kinda reminds me of this meme :
http://www.google.com/imgres?um=1&hl=en&sa=N&authuser=0&biw=1280&bih=923&tbm=isch&tbnid=LIYStgvKYShV0M:&imgrefurl=http://mozey.wordpress.com/2007/02/18/funny-exam-answers/&docid=KWLUOoQGqifSeM&imgurl=http://mozey.files.wordpress.com/2007/02/findx.gif&w=342&h=240&ei=C1aST9DBJIagiQKx2Owp&zoom=1&iact=rc&dur=389&sig=100237503914979846281&page=1&tbnh=136&tbnw=187&start=0&ndsp=31&ved=1t:429,r:0,s:0,i:70&tx=79&ty=79

Answer 18

having serious trouble drawing! It never touches the diagram where i want it to..?