yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Writing standard equation of a circle | Mathematics II | High School Math | Khan Academy


3m read
·Nov 11, 2024

  • [Voiceover] So we have a circle here and they specified some points for us. This little orangeish, or, I guess, maroonish-red point right over here is the center of the circle, and then this blue point is a point that happens to sit on the circle.

And so with that information, I want you to pause the video and see if you can figure out the equation for this circle.

Alright, let's work through this together. So let's first think about the center of the circle. And the center of the circle is just going to be the coordinates of that point. So, the x-coordinate is negative one and then the y-coordinate is one. So center is negative one comma one.

And now, let's think about what the radius of the circle is. Well, the radius is going to be the distance between the center and any point on the circle. So, for example, this distance. The distance of that line.

Let's see, I can do it thicker. A thicker version of that. This line, right over there. Something strange about my... Something strange about my pen tool. It's making that very thin. Let me do it one more time. Okay, that's better. (laughs) The distance of that line right over there, that is going to be the radius.

So how can we figure that out? Well, we can set up a right triangle and essentially use the distance formula which comes from the Pythagorean Theorem. To figure out the length of that line, so this is the radius, we could figure out a change in x.

So, if we look at our change in x right over here. Our change in x as we go from the center to this point. So this is our change in x. And then we could say that this is our change in y. That right over there is our change in y.

And so our change in x-squared plus our change in y-squared is going to be our radius squared. That comes straight out of the Pythagorean Theorem. This is a right triangle.

And so we can say that r-squared is going to be equal to our change in x-squared plus our change in y-squared. Now, what is our change in x-squared? Or, what is our change in x going to be? Our change in x is going to be equal to, well, when we go from the radius to this point over here, our x goes from negative one to six.

So you can view it as our ending x minus our starting x. So negative one minus negative, sorry, six minus negative one is equal to seven. So, let me... So, we have our change in x, this right over here, is equal to seven.

If we viewed this as the start point and this as the end point, it would be negative seven, but we really care about the absolute value in the change of x, and once you square it all becomes a positive anyway. So our change in x right over here is going to be positive seven.

And our change in y, well, we are starting at, we are starting at y is equal to one and we are going to y is equal to negative four. So it would be negative four minus one which is equal to negative five.

And so our change in y is negative five. You can view this distance right over here as the absolute value of our change in y, which of course would be the absolute value of five. But once you square it, it doesn't matter. The negative sign goes away.

And so, this is going to simplify to seven squared, change in x-squared, is 49. Change in y-squared, negative five squared, is 25. So we get r-squared, we get r-squared is equal to 49 plus 25.

So what's 49 plus 25? Let's see, that's going to be 54, or was it 74. r-squared is equal to 74. Did I do that right? Yep, 74.

And so now we can write the equation for the circle. The circle is going to be all of the points that are, well, in fact, let me write all of the, so if r-squared is equal to 74, r is equal to the square-root of 74.

And so the equation of the circle is going to be all points x comma y that are this far away from the center. And so what are those points going to be? Well, the distance is going to be x minus the x-coordinate of the center. x minus negative one squared.

Let me do that in a blue color. Minus negative one squared. Plus y minus, y minus the y-coordinate of the center. y minus one squared. Squared. Is equal, is going to be equal to r-squared. Is going to be equal to the length of the radius squared.

Well, r-squared we already know is going to be 74. 74. And then if we want to simplify it a little bit, you subtract a negative, this becomes a positive. So it simplifies to x plus one squared plus y minus one squared is equal to 74. Is equal to 74.

And, we are all done.

More Articles

View All
The Real Story of Oppenheimer
J. Robert Oppenheimer might be the most important physicist to have ever lived. He never won a Nobel Prize, but he changed the world more than most Nobel Prize winners. Under his leadership, the best physicists of the 20th century built the atomic bomb, f…
Why We’re Going Back to the Moon
That’s one small step for man, one diabetes. On July 16, 1969, Apollo 11 blasted off into space carrying three astronauts bound for the Moon. Four days later, Neil Armstrong became the first man to ever set foot on our celestial neighbor, marking a new e…
15 Reasons Why People Fail
You know, failure is something that happens to all of us at some point. It’s like running into a roadblock when you’re trying to get somewhere. What’s even more concerning is that many times we find ourselves tripping over our own shoelaces, so to speak. …
These are the most INSANE mansions of Beverly Hills 90210
Let’s see if we can clear this without scraping. Make sure to smash that like button. If this video gets three thousand likes, then I’m not gonna scrape; otherwise, if it doesn’t get three thousand likes, I’m gonna scrape my bumper pretty bad here. Oh, w…
Over 100,000 Sea Turtles Nest at the Same Time. How? | National Geographic
My main interest is understanding how, or specifically what the mechanism is for these sea turtles to synchronize their nesting behaviors. We do not know why the sea turtles specifically come to Austin. Sea turtles are renowned for their ability to trave…
Hydrodynamic Levitation!
Check this out! Hahaha, isn’t that awesome? That is hydrodynamic levitation. Check it out! This styrofoam ball is levitating on this stream of water, and it’s doing so in a very stable way. The set up is so stable you can play Frisbee through it, which is…