Consider a circle with radius 5 and another circle with radius 3. let d represent the distance between the two centers. we want to know how many intersections there are of these two circles for different values of d. draw figure id d=10?

If d<2, there is no intersection

If d=2, there is one intersection

If 2<d<8, there are two intersections

If d=8, there is one intersection

if d>8, there is no intersection

Step-by-step explanation:

If d<2 (the difference between both radius is 2), the little circle is inside the big one. Thus there is no intersection.

If d=2, the little circle is inside but tangent to the big one. There is one intersection then.

If 2<d<8, there are two intersections since the little circle has a portion outside the big one, and another portion inside.

If d=8, the little circle is tangent to the big one from outside. There is one intersection then.

if d>8, the little circle is completely exterior to the big one. Thus, there is no intersection.

Please find attached the figure for d = 10. The big circle is centered and the other is offset by 10.

the circle one