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## Precalculus (2018 edition)

### Course: Precalculus (2018 edition)>Unit 2

Lesson 2: The features of a circle

# Features of a circle from its graph

Given a circle on the coordinate plane, Sal finds its center and its radius.

## Want to join the conversation?

• instead of eye balling the center, how do you find the center if you only given a point on the circle. Trial and error?
• If you have one point, and you do not know anything else, there are an infinite amount of circle centers that could be scattered about the Cartesian plane. Even with two points, there are still an infinite amount, though in a line equidistant between the points now. You need to know three points on a circle to define it. We could also know the center point and another point on the circle, which is what we are doing here. Sal 'eye-balled' the center and was given a point on the circle, which he then used to find the radius. If we do not know the center and are given a point on the circle, we are no better off than before.
• conic sections aren't functions, right? They have more than one corresponding point for x. Yet a parabola is indeed a function... (?)
• You're right: Conic sections are a bit like trees or rocks: There are all kinds of them, and they don't all look alike, or act alike. Whether or not a particular curve (conic section or not) is a function will depend on how we write its equation, or how we situate it on a coordinate plane. The equation of a closed curve (like a circle or an ellipse) can never be a function (at least: I don't think so. Can you say why?). The equation of any parabola or hyperbola can be written as a function, but there are equations of parabolas and hyperbolas that are not functions. For example, if we think of y as a function of x, then x^2 = y is a function, but x = y^2 is not, even though the two equations have congruent graphs.
• how we find the center of a circle if it cannot be eye balled?
• If you do not have the formula of it, you need at least three points you can eyeball on the circle, not inside. If you cannot tell any points exactly then there's no way to get the center.

With the three points you have, there are a few ways to do it, this is the way i prefer. You need to find two perpendicular bisectors (I can explain this if you need.) It doesn't matter which points you do, but you need to find two different perpendicular bisectors. Then, wherever these two intersect is the center of the circle.

Let me know if that doesn't help, of again if you would like an explanation on perpendicular bisectors.
• Instead of using the distance formula couldn´t you just count up from a point on the same line on the center?
(1 vote)
• In some problems you could, but often the circle does not intersect at an exact point, so you would have to guess where it intersects. The distance formula generally gives you a more accurate answer. For example, in this video you would be able to count that the radius is about 2.25 units, but you would not be able to count that it is exactly √5 units.
• how to determine the radius if the circle has its center?
(1 vote)
• You find the distance between the center and the any point on the edge of the circle.
• What's the distance formula again? Was it sqrt(x_2-x_1)^2+(y_2-y_1)^2
(1 vote)
• I understand what you mean, but you have to square root everything.
(1 vote)
• 1) I dont get why the change in x is 1 unit to the right, why couldnt it be one unit up.
2) why is it called CHANGE in x/y nothing is changing.
3) distance is just the radius right?
Thanks
(1 vote)
• The change in x has to be to the right or left, because the x-axis is horizontal.
The "change" in y or x just means how many spaces apart the point and the center are (the number has changed).
Yes, the distance is same as the radius.
• Was the radius that we solved for, an exact value or just an approximated?
cuz the result turns out to be only the "sqrt" of some number.. not any other number
• The radius, which turned out to be √5, is an exact number when written that way; it is not an approximation. If it had been written as a decimal, it would have had to be approximated because irrational numbers in decimal form never end and never repeat.
• how to determine the radius?
• So if you know the radius, then you know that pi*(radius^2) equals the area, so if we know the area, then we could just do a little bit of rearranging of the equation, here's the steps I went through to get to an equation which if you plug in the area of a circle, you can find the radius:
Area = Pi * radius^2 || divide pi by both sides
Area/Pi = radius^2 || take the square root of both sides
sqrt(Area/Pi) = radius || and there you go, we got that the radius is equal to sqrt(Area/Pi).
I haven't tried that out myself ever, but it should be right, hope this helps,
- ncochran2