# Can a point be 5 unit of lenght from a line?

• May 13th 2011, 06:33 AM
Anna55
Can a point be 5 unit of lenght from a line?
Is there any points on the line y=2x-1 which fulfill the demand that the distance to the point (1;3) should be 5 unit of length?

I tried to draw the graph and mark the point (1;3), however I still do not know how to do the calculations. Thank you in advance!
• May 13th 2011, 06:42 AM
Ackbeet
There are three possibilities:

A. The point on the line that is closest to (1,3) is greater than five units from (1,3). In this case, there are no points satisfying your criterion.
B. The point on the line that is closest to (1,3) is exactly five units from (1,3). In this case, there is exactly one point satisfying your criterion.
C. The point on the line that is closest to (1,3) is less than five units from (1,3). In this case, there are exactly two points satisfying your criterion.

You could try to find the point on the line closest to (1,3), but I claim that's more work than you need to do. If you plot the point and the line, you can see at a glance that the point (1,1) is on the line. What is the distance from (1,1) to (1,3)? What does that say about the distance from (1,3) to the point on the line closest to (1,3)? So which case are we in?
• May 13th 2011, 06:55 AM
Plato
Quote:

Originally Posted by Anna55
Is there any points on the line y=2x-1 which fulfill the demand that the distance to the point (1;3) should be 5 unit of length?

Can you solve $(x-1)^2+([2x-1]-3)^2=25~?$
• May 13th 2011, 03:35 PM
bjhopper
Hi Anna55,
On your graph draw a circle radius 5 from center (1,3). Write a second equation for the circle

( X-h)^2 + (y-k)^2 = r^2 where h and k are the center coordinates. Solve the two equations for the the two points where the circle meets the straight line

bjh