Video Transcript
In this video, we will learn how to
find the positions of points, straight lines, and circles with respect to other
circles. We begin by recalling that we
define a circle as being a set of points in a plane that are a constant distance
from a point in the center. The line segment from the center to
a point on the circumference is called the radius, which is usually denoted with a
lowercase 𝑟.
We will begin by considering how
points can be positioned with regard to a circle. In the figure drawn, we can see
that there are three distinct possibilities in terms of where points can exist on a
plane with respect to the circle: one, inside the circle as in point 𝐴; two, on the
circle as in point 𝐵; or three, outside the circle as in point 𝐶. These distinctions matter in terms
of the distance that points can have from the center of the circle compared to the
radius.
Let’s imagine that the the three
points lie on the same straight line as shown. We can see that the distance from
the center 𝑀 to the point 𝐴 is less than the radius 𝑟. The distance from the center to
point 𝐵 is equal to the length of the radius. And the distance 𝑀𝐶 is greater
than the radius 𝑟. This can be generalized as
follows. For a circle with center 𝑀 and
radius 𝑟 and general point 𝑃, if 𝑀𝑃 is less than 𝑟, then 𝑃 is inside the
circle. If 𝑀𝑃 equals 𝑟, then 𝑃 lies on
the circle. If 𝑀𝑃 is greater than 𝑟, then 𝑃
is outside the circle. We will now consider an application
of this rule in an example.
A circle has a radius of 90
centimeters. A point lies on the circle at a
distance of three 𝑥 minus three centimeters from the center. Which of the following is
true? (A) 𝑥 is less than 31, (B) 𝑥
is equal to 31, or (C) 𝑥 is greater than 31.
Let’s begin by sketching the
circle. We are told it has a radius of
90 centimeters. We are also told that a point,
which we will call 𝑃, lies on the circle. Letting the center of the
circle be point 𝑀 and the radius 𝑟, we recall that if a point lies on a
circle, then its distance from the center is equal to the radius. So 𝑀𝑃 equals 𝑟. We know that the distance 𝑀𝑃
is equal to three 𝑥 minus three centimeters. So, three 𝑥 minus three must
be equal to 90. Adding three to both sides of
our equation, we have three 𝑥 equals 93. We can then divide through by
three such that 𝑥 is equal to 31.
So, the correct answer is
option (B). If a circle has a radius of 90
centimeters and a point 𝑃 lies on the circle at a distance of three 𝑥 minus
three centimeters from the center, then 𝑥 is equal to 31. Whilst it is not required in
this question, we note that if 𝑥 is less than 31, the point would lie inside
the circle. And if 𝑥 is greater than 31,
then the point lies outside the circle. Having seen the possible
relations between points and circles, we will now move on to consider how
straight lines relate to circles.
As before, we will begin by
considering a circle with center 𝑀. Once again, there are three
distinct possibilities in terms of how a line can intersect with a circle. Firstly, a secant to a circle
intersects the circle twice, in this example at points 𝐴 and 𝐵. Secondly, we have a tangent, which
intersects or touches the circle once. Finally, we have the situation
where a line is outside the circle completely and therefore does not intersect the
circle.
Much like with a single point, it
is possible to determine which of the classifications a line belongs to by
considering its distance from the center of the circle. In order to do this, we need to
recall the following definition. The distance from a point 𝐴 to a
line 𝐿 is the shortest distance possible from 𝐴 to any point 𝐵 that lies on
𝐿. This is equal to the length of the
perpendicular line segment that joins 𝐴 to the nearest point on the line. Using this definition, let us find
the distance between lines 𝐿 one, 𝐿 two, 𝐿 three, and the center of the circle 𝑀
in the diagram. We observe that compared to the
radius of the circle 𝑟, 𝑀𝑋 is less than 𝑟, 𝑀𝑌 is equal to 𝑟, and 𝑀𝑍 is
greater than 𝑟.
Just as we did before for single
points, we can generalize this to a rule for any line and its classification as a
secant, tangent, or line lying outside the circle. For a circle with center 𝑀 and
radius 𝑟 and a line 𝐿, where 𝐴 is the closest point to 𝑀 that lies on 𝐿, if
𝑀𝐴 is less than 𝑟, 𝐿 is a secant to the circle. If 𝑀𝐴 equals 𝑟, 𝐿 is a tangent
to the circle. If 𝑀𝐴 is greater than 𝑟, 𝐿 is
outside the circle.
Let us consider an example of an
application of the above rule.
Circle 𝑀 has radius 65. Suppose 𝐴 is on a line 𝐿 and line
segment 𝑀𝐴 is perpendicular to 𝐿. If two 𝑀𝐴 minus 56 equals 18,
what can be said of how 𝐿 lies with respect to the circle? Is it (A) 𝐿 is a secant to circle
𝑀? (B) 𝐿 is a tangent to circle
𝑀. Or (C) 𝐿 is outside of circle
𝑀.
Recall that if we compare the
distance from the center of circle 𝑀 to line 𝐿 with the radius of the circle, we
can determine whether 𝐿 is a secant, a tangent, or if it is outside the circle. In particular, the distance from 𝑀
to 𝐿 is defined by the length of the perpendicular line segment that joins 𝑀 to
𝐿. Since line segment 𝑀𝐴 is
perpendicular to 𝐿, it therefore measures the distance from 𝑀 to 𝐿. Now, 𝑀𝐴 satisfies the equation
two 𝑀𝐴 minus 56 equals 18. We can solve this to find 𝑀𝐴 by
rearranging the equation. Adding 56 to both sides, we have
two 𝑀𝐴 equals 74. We can then divide through by two
such that 𝑀𝐴 is equal to 37.
We are now in a position to compare
the length of 𝑀𝐴 to the length of the radius. Since the radius is 65 and 𝑀𝐴 is
equal to 37, we have 𝑀𝐴 is less than 𝑟. This can be illustrated as
shown. Since 𝑀𝐴 is less than 𝑟, we can
conclude that, by definition, the correct answer is option (A). That is, 𝐿 is a secant to the
circle. Note that if the length 𝑀𝐴 was
equal to the length of the radius, then line 𝐿 would be a tangent to the
circle. And if 𝑀𝐴 was greater than 𝑟,
then the line would be outside of the circle.
There is one final thing that is
worth noting when dealing with straight lines and circles. As a tangent always intersects a
circle at a single point on its circumference and this point is the closest point on
the tangent to the center, the perpendicular line segment between a tangent and the
center is always a radius of the circle. This can be more formally stated as
follows. For a circle with center 𝑀 with a
tangent 𝐿 to the circle at a point 𝐴, line segment 𝑀𝐴 is perpendicular to the
tangent and is a radius of the circle. We note that this property goes
both ways. If a line is perpendicular to a
radius of the circle and intersects that radius at a point on the circumference,
then it must be a tangent to the circle.
So far, we have seen how points can
interact with circles and how lines can interact with circles. But what about circles with other
circles? At a basic level, two distinct
circles can intersect at two points, at one point, or not at all. However, there are additional
scenarios that arise when one circle is contained within the other circle. Let us investigate these one by
one.
In all five of these cases, we
assume that we have two distinct circles, one with center 𝑀 one and radius 𝑟 one
and the other with center 𝑀 two and radius 𝑟 two. We will let 𝑀 one 𝑀 two be the
distance between the two centers. If 𝑀 one 𝑀 two is greater than 𝑟
one plus 𝑟 two, then the circles are apart from each other. They do not intersect and neither
one is inside the other. If 𝑀 one 𝑀 two is equal to 𝑟 one
plus 𝑟 two, then the circles intersect at one point, 𝐴, on the outside of both
circles. We note that there is a tangent to
both circles at 𝐴.
Suppose 𝑟 one is greater than or
equal to 𝑟 two. Then, if 𝑀 one 𝑀 two is greater
than 𝑟 one minus 𝑟 two and less than 𝑟 one plus 𝑟 two, the two circles intersect
at two points, 𝐴 and 𝐵. If 𝑟 one is greater than 𝑟 two
and 𝑀 one 𝑀 two is equal to 𝑟 one minus 𝑟 two, then the circles intersect at one
point, 𝐴, on the inside of the circle centered at 𝑀 one. Note that 𝑟 one cannot equal 𝑟
two in this case, since the circles would overlap and not be distinct. In the final case, if 𝑟 one is
greater than 𝑟 two and 𝑀 one 𝑀 two is less than 𝑟 one minus 𝑟 two, then one
circle is inside the other and the circles do not intersect.
In all five cases, the important
thing is to compare the distance 𝑀 one 𝑀 two of the centers of the circles from
one another to their radii. This can be summarized as
shown.
We will now look at an example
where we need to recall these properties.
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.
Finally, for two circles with
centers 𝑀 one and 𝑀 two and radii 𝑟 one and 𝑟 two, we saw the following. If 𝑀 one 𝑀 two is greater than 𝑟
one plus 𝑟 two, the circles are apart. If 𝑀 one 𝑀 two is equal to 𝑟 one
plus 𝑟 two, the circles touch externally. If 𝑀 one 𝑀 two is greater than 𝑟
one minus 𝑟 two and less than 𝑟 one plus 𝑟 two, the circles intersect twice. If 𝑀 one 𝑀 two is equal to 𝑟 one
minus 𝑟 two, the circles touch internally. And finally, if 𝑀 one 𝑀 two is
less than 𝑟 one minus 𝑟 two, one circle is inside the other, where 𝑟 one is
greater than or equal to 𝑟 two.
