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The Circle - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When working with circle conic sections, we can derive the equation of a circle by using coordinates and the distance formula. The equation of a circle is (x - h) ^2 + (y - k)^2 = r^2 where r is equal to the radius, and the coordinates (x,y) are equal to the circle center. The variables h and k represent horizontal or vertical shifts in the circle graph.

So as you know a circle is a infinite set of points that are equidistant from the center okay? What we're going do now is actually take a look at finding the equation for a circle.
What we're looking at is a circle with radius 4 centered on the origin okay? So what we know is that we have the origin so we have the point 0, 0 and we have a bunch of points that are 4 units we move from there okay? So we don't know what our specific point is we just know it'll be x,y and we know that the distance from the origin is 4. What we can do is use the distance formula because we're looking at a distance between a point and the origin so hopefully you remember the distance formula as distance is equal to the square root of x minus x one squared plus y minus y one quantity squared okay? So for all we have to do in this case is a distance from the origin to our point is just a radius in this case 4 and we're dealing the point we know is 0, 0 so we can just plug in 0, 0 for x one and y one and what we end up with is 4 is equal to x squared plus y squared and a giant square root of that okay x minus 0 is just the x squared is x squared, y-0 y, y squared y squared okay? So what I want to do now is get rid of this square root all I have to do is square both sides and what we end up with is 16 is equal to x squared plus y squared. Okay so that is the particular equation for this specific circle.
What I want to do now is look at the general equation okay? So for a general equation let's say we have radius r and we are centered around the point hk okay? I'd still going to go to our distance formula and so our distance in this case is going to be the distance from the center to the other side which is just r so you're going to end up with r is equal to the square root of difference of x's our first x is just a variable minus h quantity squared plus y minus k, y squared we take the square root of all that then just like we did before we want to square both sides to get rid of that square root and what we end up with is r squared is equal to quantity x minus h squared plus y minus k quantity squared. And this is the general equation for a circle okay? So we know the by looking at it we'll know the center and we'll also know the radius.

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