Two circles have the following equations: #(x -8 )^2+(y -5 )^2= 9 # and #(x +4 )^2+(y +2 )^2= 25 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?
2 Answers
Consider the line passing through the middle of the two circles. Explicit answer below.
Explanation:
We can immediately deduce that the two circles do not even overlap, let alone have one contain the other - the first has radius 3 and is centred at
To deduce in general the maximum of the distance between any two points on two curves is a variational calculus problem, which is quite an advanced technique, and not likely to be what's wanted here.
Fortunately, there's a much simpler way to get to the answer in this case -the maximum distance is found when one draws a line through the centre of the two circles and takes its far intersection with each circle. To see that this is so, consider the triangle inequality; the logic's written out here: https://math.stackexchange.com/questions/437313/maximum-distance-between-points-on-circle
So the maximum distance is the distance between the centres of the two circles plus the radii of the two circles:
As 193 is a prime number, the root does not simplify further.
The distance between the centers is
Explanation:
graph{0= ( (x-8)^2 + (y-5)^2-9) (( x + 4 )^ 2 + ( y + 2 ) ^2 - 25) [-20, 20, -10,10]}