Lesson Video: Positions of Points, Straight Lines, and Circles with respect to Circles Mathematics

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.