# Maximum Distance between points on circle

What is the greatest possible distance between two points: one on a circle with radius 1 and centre (1; 2) and the other on a circle with radius 2 and centre (4; 6)

I am not familiar with the equation of a circle, is there a way to do this without using the circle equation?

There is a way to do this without using the circle equation, yes.

Hint: start by simplifying it. Given a point and a circle, what is the greatest possible distance between that point and a point on the circle? Then how can you use that result to solve your problem?

• could i get the distance between the centres and then add the two radii? – simpleton Jul 6 '13 at 8:16
• @simpleton, indeed. But can you show that that's the correct answer? – Peter Taylor Jul 6 '13 at 8:25

Here is a sketch of a way to so this.

Let the points be $P, Q$ on circles $A, B$ with centres $O_A, O_B$, and $a,b$ be the points where the extension of the segment joining the centres meets the two circles.

Then we have $$PQ\le PO_A+O_AQ=aO_A+O_AQ\le aO_A+O_AO_B+O_BQ=aO_A+O_AO_B+O_Bb=ab$$

This uses "any two sides of a triangle are together greater than the third" for the inequalities (which may be equalities if the "triangle" is degenerate), and the fact that two radii of the same circle are equal for the first two equalities. Worth drawing a diagram.

Then you have to use the information you have about the circles to calculate the length in your specific case.