Video Transcript
Suppose we have two circles, one
with center 𝑀 one and radius 𝑟 one equals seven and one with center 𝑀 two and
radius 𝑟 two equal to four. Given that the circles intersect at
two distinct points, which of the following is the correct range of values for the
length 𝑀 one 𝑀 two? (A) 𝑀 one 𝑀 two is less than
three. (B) 𝑀 one 𝑀 two is less than
11. (C) Three is less than 𝑀 one 𝑀
two. (D) 𝑀 one 𝑀 two is grteater than
three and less than 11. Or (E) 𝑀 one 𝑀 two is greater
than four and less than seven.
We are told in this question that
two circles intersect at two distinct points. The larger circle has center 𝑀 one
and a radius of seven, whereas the smaller circle has center 𝑀 two and radius equal
to four. We are asked to find the range of
values for the length 𝑀 one 𝑀 two, which is the distance between the two
centers. We recall that when two circles
intersect at two distinct points, then 𝑀 one 𝑀 two is greater than 𝑟 one minus 𝑟
two and less than 𝑟 one plus 𝑟 two. Substituting in the values in this
question, we have 𝑀 one 𝑀 two is greater than seven minus four and less than seven
plus four. This simplifies to 𝑀 one 𝑀 two is
greater than three and less than 11. So, the correct answer is option
(D).
We will now finish this video by
summarizing the key points.
We began by considering the
relationship between a point and a circle. In this case, there were three
possible scenarios. If the distance between the center
𝑀 and a general point 𝑃 is less than the radius, then the point lies inside the
circle. If the length 𝑀𝑃 equals 𝑟, the
point lies on the circle. And if 𝑀𝑃 is greater than 𝑟,
then the point is outside the circle. Next, we looked at the relationship
between a straight line and a circle. Once again, there were three
possibilities. If 𝑀𝐴 is less than 𝑟, then 𝐿 is
a secant to the circle. If 𝑀𝐴 equals 𝑟, then 𝐿 is a
tangent to the circle. And if 𝑀𝐴 is greater than 𝑟,
then 𝐿 is outside the circle, where 𝐴 is the point on 𝐿 that is closest to the
center of the circle M.
