Geometri Analitik Fase F | Persamaan Lingkaran Bagian 1 - Konsep Dasar dan Persamaan Lingkaran

Assalamualaikum Wr Wb Meet me again, Denny Handayani on the MATLAB channel This is the first part of the video, we learn circle material In this video we will learn the basic concept of circles and the similarity of circles Either centered at 0.0 or circles that are centered at 0.0 Okay, now let's just discuss the material

Okay, before we discuss the equation of a circle, you should first understand the definition of a circle. The definition of a circle is like this. A circle is a place of a point that is the same distance from a certain point. The illustration is like this, friends.

This is what is said to be a certain point, this certain point or we usually call it a center point. The same distance, which is called distance, is the fingers of the circle. So here, this is a collection of points. So what is called a circle is this, friends. So this is what is called a circle. Circles do not have a space, because this part is not the content.

Not this one. So if the contents are here, this is not a circle. But this is the area that is limited by the circle. Once again, this is not a circle. Be careful, this is not a circle. But the circle is here. The edge part. This is called a circle. Even though it's wide, it's wide that is limited by the circle. Now let's learn the equation of circles.

We start from the circle that is centered at point O or 0.0 I will not give a formula directly, we look for it ourselves so that you can understand better This is the Cartesian diagram, we make a circle that is centered at point O, this is point O 0.0 For example, this is a circle

We draw a line of fingers, for example, this is a point, we just call it a point, the coordinate is x, y, so here the absciss is x and the coordinate is y. The distance from O to T is called the circle fingers. Okay, to make it clearer, this absciss we pull here, it means the absciss is x here, right? It means this is x and the coordinate is y.

Now, pay attention to this part, we make it a triangle. So the distance from O to X, this is the same as X, right? From X to 11 here, this is the same as from O to Y. So the distance is Y. By using Pythagoras, we can find the length of R, the length of the hypotenuse.

So, what is r? Use Pythagoras, r^2 = x^2 + y^2 Or we can write x^2 + y^2 = r^2. Now this is the circle equation that is centered at point 0.0. Easy, right? Okay, an example of the question. Determine the circle equation that is centered at point O 0.0 and through point -3.4.

When you want to find the circle equation that is centered at 0,0, what you need to find first is only one, namely the R, the fingers, or the R squared. In this question, the R is not there, we have to find it. The way you guys substitute this equation, replace the X with negative 3, replace the Y with 4.

The circle of x squared plus y squared equals r squared. We replace x with negative 3. So negative 3 squared plus y replaced by 4. 4 squared equals r squared. Negative 3 squared 9 plus 4 squared 16 equals r squared. 9 plus 16.

25, 25 equals R squared or R squared equals 25. Well, you don't have to look for R, you don't have to be R equals 5. Why? Because what we need is R squared. But if you want, it's okay, we know the fingers are 5.

So the circle equation is x^2 + y^2 = r^2, and we replace the r^2 with 25. So the circle equation is x^2 + y^2 = 25. Done. Now we will look for the circle equation that is centered at other points other than 0.0, for example at points P, A, B.

the illustration, pay attention, this is the Cartesian diagram, let's say a circle here, this is the center at A, B, which means the absciss is A, the center coordinate is B

As usual, we draw a finger line, then this point we divide by the point T , which means the absciss is X and the ordinate is Y. From P to T, this is called the circle fingers. We make another triangle like the previous one, so that we can easily use Pythagoras. Now, friends, pay attention.

From here to here, what is the distance? It's the same as from here to here, right? It means that X is minus A, right? This is X minus A. From here to here, what is the distance? It means that Y is minus B. I write Y minus B from here to here. Now we can find the length of the R using Pythagoras.

R squared equals X minus A squared plus Y minus B squared. Or we can write X minus A squared plus Y minus B squared equals R squared. Well, this is the circle equation centered on A comma B. So the center point is A comma B and the fingers are R. Now let's discuss an example.

Determine the circle equation that is centered at point P and has four fingers. We just substitute it. This is the center point, 2, -3 means that this 2 is the a, -3 is the b, and has four fingers, this is the r. We just substitute it to this equation.

x-a^2 + y-b^2 = r^2 we replace a with 2, we replace b with -3 and we replace r with 4 so we get x-2^2 + y-3^2 = 4^2

So, x-2 squared, y-3, y-3 means it's an addition. Y + 3 squared = 4 squared. Oh yes, we just replace 4 squared with 16. And this is the equation of the circle.

How if you want to simplify again, x-2 is squared, so x squared is -4x + 4y + 3 is squared again, that's okay. And later we will get the general form. Meanwhile, let's stop here first. Now let's discuss the general form of the circle equation. Okay, the general form of the circle equation.

This is the circle equation that is centered at . If it is centered at , we don't need to do anything else, it's already the general form.

Now we will change this shape. x-a squared plus y-b squared equals r squared. x-a squared, so x squared minus 2ax plus a squared. Then add, we square y-b, so y squared minus 2b plus b squared equals r squared. Now this position we

change it, so x squared first then add y squared minus 2ax minus 2by like that then add a squared plus b squared minus r squared equals zero, now we simplify the writing here minus 2a minus 2b and a squared plus b squared minus r squared, we say this is another variable, friends, because this is a value, right? we change this

Minus 2a is for example a, minus 2b is for example b, and a squared plus b squared minus r squared is for example c. So the general form of the circle is x squared plus y squared plus ax plus by plus c equals 0. This is more simple.

This is the general form of the circle equation. With A, A capital is negative 2A. Where A non-capital is the abscissus coordinate of the center point. So if you want to find the abscissus of the center point, it means that this A capital, this coefficient X, you just have to divide it by negative 2. Once again, the abscissus of the center point is A divided by negative 2. Then B.

b = -2b, if you want to find the coordinate of the center point or the small b, it's b divided by -2. Then from here we get c is a^2 + b^2 - r^2. If you want to find r, r^2 = a^2 + b^2 - c or r = root of a^2 + b^2 - c, this is the r. Example question.

Determine the center point coordinates and fingers from the following circle. This is the circle equation that has been written in the general form. We will look for the center point coordinates and fingers from this circle. The way is easy.

To find the center point, the abscissus and the ordinate, a and b are each x coefficients, the x coefficients are a, we divide by -2, and the y coefficients are b, we divide by -2. So to find the center point, you just divide the x coefficients, which are here, and the y coefficients, we divide by -2.

-2 divided by -2 is 1, then 4 divided by -2 is -2. So we get the center point coordinate. Or for example, P = 1,-2. Now we look for the fingers.

We use this formula, friends. Where a and b are the coordinates of the center point. So this one is a, negative 2, this is b. So if you want to find r, r is equal to root a squared plus b squared minus c. a squared plus b squared minus c. Where is c? c is this 11. Negative 20. We just enter the value.

a is 1, 1 squared plus b is -2 squared, minus c, c is -20. 1 squared is 1, -2 squared is 4, minus -20, so plus 20. The same as the root of 25. We already got the center and the fingers.

So the coordinate of the center of the circle is 1,-2 and the fingers are 5. Okay, now I will give 10 exercises, please try it. Inshaallah the discussion of the subject will be discussed in the next video.

Okay, that's it for this video, see you in the next video. Assalamualaikum warahmatullahi wabarakatuh.